riemann surfaces

Riemann surfaces

Pablo Ares Gastesi

School of Mathematics, Tata Institute of Fundamental Research,

Bombay 400 005, India

[email protected]

Acknowledgements

I would like first of all to thank V. Srinivas for his continuous encouragement to

write this book.

I would also like to thank the people who attended my course, on which these notes

are based: Pralay Chatterjee, Preeti Raman and Vijaylaxmi Trivedi.

Finally I would like to thank R.R. Simha for many comments, all of them very

helpful.

Contents

Acknowledgements iv

Chapter 1. Riemann surfaces 1

1.1. Background 2

1.2. Topology of Compact Orientable Surfaces 12

1.3. Riemann Surfaces and Holomorphic Mappings 17

1.4. Differential Forms 28

1.5. Sheaf Cohomology 42

Chapter 2. Compact Riemann surfaces 57

2.1. Divisors 58

2.2. Dolbeault’s Lemma and Finiteness Results 62

2.3. The Riemann-Roch Theorem 68

2.4. Line bundles and Divisors 71

2.5. Serre Duality 81

2.6. Applications of the Riemann-Roch Theorem 95

2.7. Projective embeddings 99

2.8. Weierstrass Points and Hyperelliptic Surfaces 106

2.9. Jacobian Varieties of Riemann Surfaces 115

Chapter 3. Uniformization of Riemann surfaces 127

3.1. The Dirichlet Problem on Riemann surfaces 128

3.2. Uniformization of simply connected Riemann surfaces 141

3.3. Uniformization of Riemann surfaces and Kleinian groups 148

3.4. Hyperbolic Geometry, Fuchsian Groups and Hurwitz’s Theorem 162

3.5. Moduli spaces 178

Exercises 187

v

vi CONTENTS

Bibliography 201

Notation 203

Index 206

List of Figures 213

CHAPTER 1

Riemann surfaces

1.1 Background 2

1.2 Topology of Compact Surfaces 12

1.3 Riemann Surfaces and Holomorphic Mappings 17

1.4 Differential Forms 28

1.5 Sheaf Cohomology 42

1

2 1. RIEMANN SURFACES

1.1. Background

This section is divided into three parts. In the first one (§§1.1.1 to 1.1.13) we

review some results of Complex Analysis that we need in this book. Some of those

results will be extended later to the setting of Riemann surfaces. We also introduce

a few results from the theory of normal families of holomorphic functions, a topic

that might not be very familiar to some readers.

The second part (§§1.1.14 to 1.1.23) contains basic concepts and results of Topol-

ogy: fundamental groups, covering spaces and partitions of unity.

In the last part (§§1.1.24 and 1.1.25) we state a theorem of Schwartz on operators

on Banach spaces that we need in section 2.1.

References for the results in this section are given in §1.1.26.

Complex Analysis

1.1.1. Definition. Let Ω be an open subset of the complex plane. A complex

valued function f : Ω → C is holomorphic at a point z0 ∈ Ω if the complex

derivative

f ′(z0) = limz→z0

f(z) − f(z0)

z − z0

exists. We say that f is holomorphic in Ω if it is holomorphic at every point of Ω.

1.1.2. For a complex valued function f with partial derivatives, ∂f/∂x and

∂f/∂y, we define the complex derivatives fz and fz by the expressions (see also 1.4.1),

fz =∂f

∂z=

1

2

(∂f

∂x− i

∂f

∂y

), fz =

∂f

∂z=

1

2

(∂f

∂x+ i

∂f

∂y

).

Informally speaking, the next result says that a function is holomorphic if “it depends

on z but not on z”.

Proposition. Let f : Ω → C be a function with continuous partial derivatives

at all points of Ω. Then f is holomorphic if and only if fz = 0 (in Ω). If f is given

by f(z) = u(z) + iv(z), where u and v are real valued functions defined on Ω, then

1.1. BACKGROUND 3

f is holomorphic if and only if it satisfies the Cauchy-Riemann equations:

∂u

∂x=∂v

∂y,

∂u

∂y= −∂v

∂x.

The functions u and v above are called the real and imaginary parts of f . Sometimes

we will use the notation Re(f) and Im(f) for u and v respectively.

1.1.3. We record for later use (2.2.4) how the operators ∂∂z

and ∂∂z

behave

under composition [18, pg. 31]. Assume f : Ω → C and g : Ω′ → C are holomorphic

functions with f(Ω) ⊂ Ω′. Then

(g f)z = (gz f) (fz) + (gz f) (fz)

(g f)z = (wz f) (fz) + (gz f) (fz).

1.1.4. Proposition. A function f : Ω → C is holomorphic if and only if

for every point z0 ∈ Ω, there exists a neighbourhood U ⊂ Ω of z0, such that f can

be written as a convergent power series f(z) =∑∞

n=0 an (z − z0)n in U . If f is

holomorphic, then f ′(z) =∑∞

n=1 an n (z − z0)n−1.

Corollary. Holomorphic functions are smooth (of class C∞) that is, they have

derivatives of all orders.

We actually have that holomorphic functions are of class Cω; that is, they are

representable by power series (there exist functions with derivatives of all order

which are not given by power series).

The following result simplifies the local expression of holomorphic functions.

1.1.5. Proposition. If f : Ω → C is a non-constant holomorphic function,

then for every point z0 ∈ Ω, there exists a positive integer n, and a neighbourhood

U of z0 in Ω, such that f(z) = (z − z0)ng(z), where g : U → C is a holomorphic

function satisfying g(z0) 6= 0.

The integer n is called the order of the zero of f(z) − f(z0) at z0.


1.1.6. Theorem (Identity Principle). Let f, g : Ω → C be two holomorphic

functions defined on a connected, open subset Ω of C. Assume that there exists a

sequence znn of points of Ω, with zn →nz0, where z0 ∈ Ω, such that f(zn) = g(zn),

for all n. Then f(z) = g(z) for all z ∈ Ω.

Remark. There are two important observations to be made about the Identity

Principle: first of all, the set Ω is assumed to be connected; otherwise, the result

will hold only in the connected component of Ω containing the sequence znn and

the limit point z0. Secondly, the limit point z0 must belong to the set Ω; there exist

examples of distinct holomorphic functions that take the same values at each point

of a convergent sequence of points of an open set, where the limit point of such

sequence does not belong to the open set.

Recall that a subset A of a topological space X is called discrete if A does not have

accumulation points in X (A′ = ∅).Corollary. Let f : Ω → C be a non-constant holomorphic function. Then for

any complex number a, the set z ∈ Ω; f(z) = a is discrete in Ω (it does not have

limit points in Ω).

1.1.7. Denote by D the unit disc, D = z ∈ C; |z| < 1.Lemma (Schwarz). Let f : D → C be a holomorphic function satisfying f(0) = 0

and |f(z)| ≤ 1. Then

(1) |f(z)| ≤ |z|;(2) |f ′(0)| ≤ 1.

Moreover, if equality holds in (1) for a point z 6= 0, or in (2), then f(z) = λz, for a

complex number λ satisfying |λ| = 1.

1.1.8. A point z0 ∈ C is called an isolated singularity for a function f if f

is defined and holomorphic on U\z0, where U is a neighbourhood of z0. Isolated

singularities are of three types:

(1) removable if f can be extended to a holomorphic function in U ;

(2) pole if there exists a positive integer n, such that (z − z0)nf(z) can be

extended to a holomorphic function in U ;

(3) essential singularity if none of the above conditions are satisfied.

1.1. BACKGROUND 5

The following result describes the conditions under which a singularity is removable.

Theorem (Removable Singularity Theorem). An isolated singularity z0 of the

function f is removable if and only if there exists a neighbourhood V of z0, such that

f is bounded in V \z0. Equivalently

limz→z0

(z − z0)f(z) = 0

in V \z0.

Suppose z0 is a pole of f and let n be the smallest (positive) integer such that

(z − z0)nf(z) = g(z) is holomorphic on U . The function g(z) has a power series

expansion in U ; dividing by (z − z0)n we get that f has the following series expression

(called Laurent series), valid in U\z0,

f(z) =a−n

(z − z0)n + · · · + a−1

z − z0+

∞∑

j=0

aj(z − z0)j,

with a−n 6= 0. The point z0 is called a pole of order n; the coefficient a−1 is called

the residue of f at z0.

Definition. A function f is said to be meromorphic in a domain Ω if there

exists a discrete subset A ⊂ Ω such that f is holomorphic in Ω\A, and has a pole

at each point of A.

1.1.9. The following two results are classical and important properties of holo-

morphic functions that we will use frequently. In these results, we let Ω denote a

connected open subset of C.

Theorem (Maximum Modulus Theorem). Let f : Ω → C be a holomorphic

function. Assume that there exists a point z0 ∈ Ω such that |f(z)| ≤ |f(z0)| for all

z ∈ Ω. Then f is constant.

An easy consequence of this theorem is that if f is holomorphic on Ω and K is a

compact subset of Ω, then the maximum of |f | on K is given by the value of f at a

point in the boundary of K.

Theorem (Open Mapping Theorem). Let f : Ω → C be a non-constant holo-

morphic function. If U ⊂ Ω is an open set, then f(U) is an open set (in C).


1.1.10. The next theorem tells us that the inverse of a bijective, holomorphic

mapping is also holomorphic. Observe that this result is not true in Real Analysis;

for example, the function f : R → R given by f(x) = x3 is bijective and infinitely

differentiable, but the inverse function does not have derivative at the point x = 0.

Theorem. Let f : Ω → C be a one-to-one holomorphic function. Then f ′(z) 6=0 for all z ∈ Ω and f−1 : f(Ω) → C is holomorphic.

1.1.11. The following local description of holomorphic functions (see [22, The-

orem 10.32, pg. 216]) is an easy consequence of 1.1.5. Its generalisation to Riemann

surfaces (1.3.10) will be a very useful tool.

Proposition. Let f : Ω → C be holomorphic. Let z0 ∈ Ω, and n the order of the

zero of f(z)−f(z0) at z0. Then there exists a neighbourhood U of z0 in Ω, a positive

number r, and a holomorphic function g : U → C, such that f(z) − f(z0) = g(z)n

in U . The function g satisfies |g(z)| < r and g′(z) 6= 0, for all z ∈ U .

1.1.12. A sequence of holomorphic functions fnn, defined on a domain (or

an open set) of C, is said to converge uniformly on compact subsets of Ω to a

function f , if for every compact set K ⊂ Ω, the sequence fnn converges uniformly

to f in K; that is, for every compact subset K of Ω and every ǫ > 0, there exists an

n0 such that |fn(z) − f(z)| < ǫ, for all n ≥ n0 and all z ∈ K.

Theorem (Weierstrass). Let fnn be a sequence of holomorphic functions de-

fined on a domain Ω of C. If fnn converges uniformly on compact subsets of Ω

to a function f , then f is holomorphic. Moreover, the sequence of derivatives f ′nn

converges uniformly on compact subsets of Ω to f ′.

1.1.13. Definition. A family A of holomorphic functions defined on a do-

main Ω of C is called normal if for every sequence fnn of elements of A, there

exists a subsequence fnjj

that converges uniformly on compact subsets of Ω.

Observe that in the above definition we do not require that the limit functions (of

subsequences of elements of A) belong the family A. For example, if Ω = D and

A = fn(z) = z/n; n ∈ Z, then A is normal, but the limit function, f ≡ 0, is not

1.1. BACKGROUND 7

in A.

Theorem (Montel). A family of holomorphic functions on a domain Ω is nor-

mal if and only if it is uniformly bounded on compact subsets. That is, for every

K ⊂ Ω, compact, there exists a positive number M , such that |f(z)| ≤ M , for all

z ∈ K and all f ∈ A.

Since holomorphic functions are continuous, in the hypothesis of Montel’s theo-

rem we have that for each function f ∈ A there exists a constant Mf,K such that

|f(z)| ≤Mf,K , for every z ∈ K. This constant depends on f and K. The important

point about Montel’s theorem is that the family A is normal if and only if we can

take a constant M for all functions on A.

Topology

1.1.14. Definition. Let n be a positive integer. A manifold M of dimension

n is a Hausdorff topological space where every point has a neighbourhood homeo-

morphic to an open ball in Rn. A manifold of dimension 2 is called a surface.

Some books require manifolds to be second countable (have a countable basis of

open sets); that is, there exists a countable collection of open sets, say A = Uj∞j=1,

such that any open subset V of M can be written as a union of elements of A.

Riemann surfaces always satisfy this condition; see remark 4 in 1.3.3.

1.1.15. For the rest of this section all topological spaces (manifolds and sur-

faces in particular) are assumed to be connected, unless otherwise stated.

Definition. A surjective continuous map between two (connected) topological

spaces p : X → Y is called a covering map if for every point q ∈ Y there is a

neighbourhood V ⊂ Y , such that p−1(V ) consists of a disjoint collection of open

subsets Uj ⊂ X, p−1(V ) =⊔j∈J Uj , and the restriction of p to each of these sets,

p|Uj: Uj → V , is a homeomorphism. The set V is called an evenly covered neigh-

bourhood of q.

By an abuse of notation we will say that X is a covering space of Y (although the


map p is part of the definition). One can easily check that X is a manifold if and

only if Y is a manifold (exercise 7).

1.1.16. A path (or curve) on a topological space X is a continuous map

c : [a, b] → X, from an compact interval of R to X. Although a path is actually a

mapping, sometimes in an abuse of notation we will identify a path with its image

c(t); a ≤ t ≤ b. We will usually consider paths defined on the interval [0, 1].

A space is said to be path connected if for any two points x0 and x1, there exists

a path c satisfying c(0) = x0 and c(1) = x1. We say that X is locally path

connected if every point has a path connected neighbourhood. It is easy to check

that a connected, locally path connected space is path connected (exercise 6). In

particular, connected manifolds are path connected, since open balls in Euclidean

space are path connected (the segment between two points in a ball is contained in

the ball).

The next proposition tells us that a path on a space can be lifted to a covering space;

we will see a more general result in 1.1.21.

Proposition. Let p : X → Y be a covering map, and c : [0, 1] → Y a path.

Then there exists a path c : [0, 1] → X, such that p c = c. Moreover, if c(0) = y0,

and x0 is a point in X with p(x0) = y0, then c is uniquely determined by requiring

that c(0) = x0.

1.1.17. For the rest of this section we will assume that all topological spaces

under consideration are path connected.

Definition. Let c1, c2 : [0, 1] → X be two paths on a topological space X with

the same initial and end points, c1(0) = c2(0) = p0 and c1(1) = c2(1) = p1. We say

that c1 and c2 are homotopic if there exists a continuous mapH : [0, 1]×[0, 1] → X,

satisfying the following properties:

(1) H(t, 0) = c1(t), for all t ∈ [0, 1];

(2) H(t, 1) = c2(t), for all t ∈ [0, 1];

(3) H(0, s) = p0, for all s ∈ [0, 1].

(4) H(1, s) = p1, for all s ∈ [0, 1].

1.1. BACKGROUND 9

We will write c1 ∼ c2 to denote that these two paths are homotopic. See figure 1.

c1

cs

c2

p0

p1

Figure 1. Homotopy between two paths.

1.1.18. A loop (based) at a point x0 ∈ X is a path c with c(0) = c(1) = x0.

Let Ω(X, x0) denote the set of loops at x0 in X. It is not difficult to see that ∼ is

an equivalence relation in Ω(X, x0).

Given two loops based at x0, say cj : [0, 1] → X, j = 1, 2, we define the composition

c1 · c2 of c1 and c2 as the path given by the following equation:

(c1 · c2)(t) =

c1(2t), 0 ≤ t ≤ 1

2,

c2(2t− 1), 12≤ t ≤ 1.

Informally speaking, c1 · c2 is the path that “first runs c1 and then c2”. It is not

difficult to see that the composition is preserved under homotopy; more precisely, if

c1 ∼ c′1 and c2 ∼ c′2, then (c1 · c2) ∼ (c′1 · c′2). It turns out that the quotient space

Ω(X, x0)/ ∼ with the composition is a group, called the fundamental group of X

at x0, written as π1(X, x0). The identity is given by the class of the constant path

cx0(t) = x0. For c1 as above, let c−11 denote the path given by c−1

1 (t) = c1(1 − t);

then we have that c1 · c−11 is homotopic to the constant path cx0; that is, the class

of c−11 is the inverse of the class of c1 (see [16, chapter 8] for details).

1.1.19. Given two points x0 and x1 in a path connected space, the fundamental

groups π1(X, x0) and π1(X, x1) are isomorphic; thus we will speak of the fundamental

group of a space, when reference to the base point is not important.

A space is said to be simply connected if its fundamental group is trivial. A

manifold has a simply connected covering, called the universal covering space [16,

pg. 397, Corollary 14.5.b]. Any two universal covering spaces are homeomorphic.

Let f : X → Y be a continuous map, x0 ∈ X and y0 = f(x0). If c is a loop

based at x0, the composition f c is a loop based at y0. Moreover, if c1 ∼ c2, then


(f c1) ∼ (f c2). The mapping f∗ : π1(X, x0) → π1(Y, y0), given by f∗([c]) = [f c]is a group homomorphism (we use square brackets for the classes of the loops in

the corresponding fundamental groups). If p : X → Y is a covering map, the

homomorphism p∗ is injective ( exercise 12). Moreover, any subgroup of π1(Y, y0)

is realised as the image of the fundamental group π1(X, x0) of some covering space

X (see [13] for more details on covering spaces).

1.1.20. Definition. Let p : X → Y be a covering map. A covering or deck

transformation is a homeomorphism f : X → X, such that p f = p. The set of all

deck transformations will be denoted by Deck(X/Y ).

In the case of the universal covering (X simply connected), the groups Deck(X/Y )

and π1(Y, y0) are isomorphic (for any point y0 of Y ). Moreover, we can “recover”

Y from its universal covering X and the group Deck(X/Y ): we have that Y is

isomorphic to the quotient X/Deck(X/Y ) [9, pg. 25, exercise 5.1].

1.1.21. The following result generalises the lifting property of paths (1.1.16).

Proposition. Let p : X → Y be the universal covering of Y . Let Z be a simply

connected topological space, and f : Z → Y a continuous map. Then there exists a

continuous map f : Z → X, such that p f = f .

1.1.22. Let p : X → Y be a covering map between two surfaces. Choose

a point y0 in Y , and an evenly covered neighbourhood V of y0. Then we have

that p−1(V ) = ⊔jUj , where p : Uj → V is a homeomorphism (for all sets Uj).

Fix j; shrinking V (and Uj) if necessary, we can find homeomorphisms φ : V →D, ψ : Uj → D, where D denotes the unit disc in the complex plane, such that

(ψ p φ)(z) = z. This leads us naturally to the concept of branched covering:

we simply require that (ψpφ(z)) = zn, for some positive integer n. This definition

is equivalent to require that every point x0 ∈ X has a neighbourhood U , such that

p : U\x0 → V \p(x0) is a covering space (see [2, I.20B, pg. 39]). Branched

coverings share many of the properties of “standard” coverings. For example, if

p : X → Y is a branched covering and c : [0, 1] → Y is a path, then there exists

a lift of c to X; that is, a path c : [0, 1] → X such that p c = c. To prove this

1.1. BACKGROUND 11

observe that in the case of X and Y being the unit disc and p(z) = zn the statement

is clearly true; the general case can be reduced locally to this particular case.

1.1.23. Let U = Uj ; j ∈ J be a collection of open sets in a manifold X.

A partition of unity subordinate to U is a collection of continuous functions,

φj : X → [0,+∞), j ∈ J , such that:

(1) supp(φj) = x ∈ X; φj(x) 6= 0 ⊆ Uj ;

(2) for every point x ∈ X, there exists a neighbourhood V such that V ∩supp(φj) 6= ∅ for only finitely many j;

(3)∑

j∈J φj(x) = 1.

Observe that by (2) the expressions in (3) are actually finite sums.

By the theorem below we have that partitions of unity exist on connected manifolds.

That situation is enough for our applications; the reader interested on more general

results can find them in a book on Topology (see the references at the end of this

section).

Theorem. Let M be a second countable, connected manifold, and U = Ujj∈Jan open covering of M . Then there exists a partition of unity subordinate to U .

A Theorem of Functional Analysis

1.1.24. Let L be a vector space over the complex numbers, not necessarily of

finite dimension. A norm || · || on L is a mapping || · || : L → [0,+∞), satisfying

the following conditions:

(1) ||v|| ≥ 0, for all v in L, and ||v|| = 0 if and only if v = 0;

(2) for v and w in L, ||v + w|| ≤ ||v||+ ||w|| (triangle inequality);

(3) for v ∈ L and λ ∈ C, ||λ v|| = |λ| ||v||.A norm induces a topology on L in a natural way; the basis of this topology is given

by the open balls

B(v, r) = w ∈ L; ||v − w|| < r,


where v ∈ L and r is a positive number (the open sets in this topologies are the

unions of open balls). A Banach space is a complete (every Cauchy sequence

converges) normed vector space.

1.1.25. A linear mapping T : L → M between Banach spaces is continuous if

and only if there exists a non-negative constant C, such that ||T (v)||M ≤ C ||V ||L(here || · ||M and || · ||L denote the norms in M and L respectively). We say that a

linear mapping is compact if there exists a neighbourhood U of 0 in L, such that

T (U) is relatively compact in M (the closure of T (U) is compact). In particular,

if T is compact and vnn is a sequence in L, with vn → 0, then there exists a

subsequence vnjj, such that T (vnj

) converges to 0 in M .

The following result will be used in 2.2.10.

Theorem (L. Schwartz). Let T, S : L → M be two continuous linear mappings

between Banach spaces. Assume that T is surjective and S compact. Then the space

(T − S)(L) has finite codimension in M .

Recall that the codimension of a subspace N in M is the dimension of the quotient

space M/N .

1.1.26. Some references for Complex Analysis are [1], [5] and [20]. For the

Topology part see [13] and [16] (Indian version of [17]). Schwartz’s theorem can be

found in [21] or [6].

1.2. Topology of Compact Orientable Surfaces

In this section we recall some facts of the Topology of compact orientable sur-

faces: genus, construction of homology groups and Euler-Poincare characteristic.

Throughout this section X will denote a connected, compact surface.

A good reference for the proofs of the results of this section is [13].

1.2.1. From the point of view of Topology compact orientable surfaces are

classified by a non-negative integer called the genus (genera in plural). Two surfaces

X and Y , of genera g and g′ respectively, are homeomorphic if and only if g = g′.

1.2. TOPOLOGY OF COMPACT ORIENTABLE SURFACES 13

A surface of genus 0 is homeomorphic to the 2-sphere,

S2 = (t1, t2, t3) ∈ R3; (t1)2 + (t2)

2 + (t3)2 = 1;

this surface is simply connected. If g > 0, X is homeomorphic to a sphere with g

handles attached (figure 2 for an example of a surface of genus 2). Its fundamental

group has a presentation given by 2g generators, a1, . . . , ag, b1, . . . , bg, and one single

relation∏g

j=1[aj, bj ] = 1, where [aj , bj] is the commutator of aj and bj ; that is

[aj , bj] = ajbja−1j b−1

j . If X has genus 1, its fundamental group is abelian; however,

if g ≥ 2 the group π1(X, x0) is not abelian, we will prove this fact in 3.3.15.

a1

b1

a2

b2

Figure 2. A surface of genus 2 with generators of the fundamental group.

1.2.2. In the next paragraphs we define the singular homology groups of com-

pact surfaces and state some results that we will need later in the text. We do

not provide proofs for those results; the reader can find more details in [2] (for the

particular case of Riemann surfaces) or [14] (for the general case). Let ∆ denote

the triangle (x, y) ∈ R2; 0 ≤ y ≤ x ≤ 1 in R2. A 2-simplex on X is a contin-

uous mapping f : ∆ → X. A 1-simplex is a continuous mapping e : [0, 1] → X.

A point, or a mapping from a point in R to X, is called a 0-simplex. The chain

groups Cn(X), n = 0, 1, 2, consists of finite formal sums of the form∑m

j=0 cjsj , where

cj ∈ Z, and cj are n-simplices (more precisely, Cn(X) is the free abelian group on

n-simplices in X). For example, an element of C0(X) is a finite sum of the type

c0 p0 + · · · + cn pn, for some points pj of X and integers cj. The sets Cn(X) are

abelian groups with “formal” addition as the group operation.

Like in the case of paths, simplices are mappings; thus two simplices might be dif-

ferent, although their images are the same: for example, if e is a 1-simplex, the

mapping e(t) = e(1 − t) : [0, 1] → X defines a simplex with the same image but a

different element on C1(X).


1.2.3. The boundary operators between chain groups, δ : Cn(X) →Cn−1(X), n = 1, 2, are defined as follows. In the case of n = 2, for a simplex

f : ∆ → X, we set δf = e0 − e1 + e2, where ej : [0, 1] → X are the simplices given

by e0(t) = f(t, 0), e1(t) = f(1, 1− t) and e2(t) = f(1− t, 1− t). Thus the boundary

of f consists of the restriction of the mapping f to the boundary of ∆ “in certain

order”. For a 1-simplex e : [0, 1] → X, we define δ(e) = e(1)− e(0). We extend δ to

C2(X) and C1(X) in the natural way to make it a group homomorphism. It is not

difficult to see that (δ δ)(c) = 0 for c ∈ C2(X); one needs to check this equality

only on 2-simplices, say f : ∆ → X,

δ(δ(f)) = δ(e1 − e2 + e3) = e1(1) − e1(0) − e2(1) + e2(0) + e3(1) − e3(0) =

= f(1, 0) − f(0, 0) − f(1, 0) + f(1, 1) + f(0, 0) − f(1, 1) = 0.

1.2.4. Let Z1(X) be the kernel of δ : C1(X) → C0(X), and B1(X) the image of

C2(X) in C1(X). The elements of these groups are called cycles and boundaries,

respectively. Since δ δ = 0, B1(X) is a (normal) subgroup of Z1(X); the quotient

group

H1(X,Z) = Z1(X)/B1(X)

is called the 1st homology group ofX (with integer coefficients). Observe that any

loop determines a 1-simplex in X without boundary, hence an element of H1(X,Z).

Thus we obtain a mapping Ω(X, x0) → H1(X,Z). We have that this mapping can be

used to compute the 1st homology group, as stated in Hurwitz’s theorem: H1(X,Z)

is isomorphic to the abelianisation of the π1(X, x0),

H1(X,Z) = π1(X, x0)/[π1(X, x0), π1(X, x0)].

Recall that ifG is a group, [G,G] denotes the subgroup generated by all commutators

of elements of G; that is, [G,G] is generated by all elements of the form aba−1b−1

for a and b in G. It follows from Hurwitz’s theorem that for a compact surface X

of genus g the group H1(X,Z) is a free abelian group of rank 2g:

H1(X,Z) ∼= Z2g = Z ⊕ 2g factors· · · · · · ⊕ Z.

1.2. TOPOLOGY OF COMPACT ORIENTABLE SURFACES 15

The homology classes of the paths in figure 2 are generators for this group. Observe

that in the case of genus 1 the fundamental group and the 1st homology group

coincide (observe that π1(X, x0) is an abelian group).

1.2.5. The 2nd homology group is defined by

H2(X,Z) = ker (δ : C2(X) → C1(X)) .

For compact orientable surfaces we have H2(X,Z) ∼= Z.

The 0th homology group, H0(X,Z), is the image of C1(X) in C0(X) under

δ. This group is easy to compute: fix a point x0 ∈ X; since X is path connected,

for any point x1 of X there exists a path c : [0, 1] → X satisfying c(0) = x0 and

c(1) = x1. Therefore δ(c) = x1 −x0. This means that the classes of these two points

in H0(X,Z) are equal; it is then easy to show that H0(X,Z) ∼= Z. In general, for

a non-connected surface Y the group H0(Y,Z) is a free abelian group of rank equal

to the number of connected components of Y .

1.2.6. As we have seen above the homology groupsHm(X,Z), form = 0, 1, 2, of

a compact surface X are free abelian groups. The rank of the nth group is called the

nth Betti number of X, denoted by bn(X). The Euler-Poincare characteristic

of X is defined as

χ(X) = b2(X) − b1(X) + b0(X).

By the above computations we have that χ(X) = 2 − 2g.

1.2.7. A triangle T on X is a set homeomorphic to the standard triangle ∆

defined above. The edges of T are the images of the edges of ∆ (under the given

homeomorphism). Similarly the images of the vertices of ∆ (the points (0, 0), (1, 0)

and (1, 1) in R2) are called the vertices of T . Observe that in this definition a trian-

gle T is a subset of X, and thus the homeomorphism between T and ∆ does not

play an important role as in the case of simplices or paths.

Definition. A triangulation of a compact surface X is a finite collection of

triangles, T = T1, . . . , Tn, satisfying the following properties:

(1)⋃nj=1 Tj = X;

(2) if the intersection of two triangles is not empty, then it is equal to either a


edge or a vertex of both triangles (see figure 3);

(3) for any vertex v, the union of all triangles containing v contains a neighbour-

hood of v.

Allowed in a triangulation Not allowed in a triangulation

Figure 3. Allowed and wrong triangulations.

T1

T2

v

Tn

Figure 4. Condition (3) in definition 1.2.7.

1.2.8. The following result gives us an alternative way of computing the Euler-

Poincare characteristic of X.

Proposition. Let T be a triangulation of a compact surface X with F triangles,

E edges and V vertices. Then

χ(X) = F − E + V.

For an example, consider the triangulation of the 2-sphere given in figure 5. We

have F = 8, E = 12 and V = 6, so χ(S2) = 2.

1.3. RIEMANN SURFACES AND HOLOMORPHIC MAPPINGS 17

v5

v4

v6

v1

v2

v3

Figure 5. A triangulation of the sphere.

1.3. Riemann Surfaces and Holomorphic

Mappings

After reviewing the necessary background in the last two sections we are ready to

introduce the concept of Riemann surface. In an informal way, a Riemann surface is

the most general topological space on which one can do Complex Analysis similar to

the analysis done in the complex plane [2]. In this section we give the formal defini-

tion of Riemann surface and some examples. We also define holomorphic mappings

between Riemann surfaces and study some of their properties. In particular we are

interested on finding which properties of holomorphic mappings (or “holomorphic”

properties in general) are satisfied by mappings between surfaces.

Riemann surfaces

1.3.1. Informally speaking one can define a Riemann surface as a collection

of open subsets of the complex plane “glued” by holomorphic functions. More pre-

cisely, let X be a connected surface; thus every point of X has a neighbourhood U

homeomorphic to an open set of R2, which we identify with C. One could define a

Riemann surface then as a (topological) surface with a covering by open sets, homeo-

morphic to open sets in C, such that if U and V are two elements of the covering with

non-empty intersection, and φ : U → φ(U) ⊂ C and ψ : V → ψ(V ) ⊂ C the cor-

responding homeomorphisms, the mapping ψ φ−1 is holomorphic. Although from

a working point of view this definition suffices (see remark after definition 1.3.2),


there are some technical problems with it. For example, the complex plane with the

identity function z 7→ z is a Riemann surface. If we now consider C with two open

sets, say U = C and V = C\0, and homeomorphisms given also by the identity on

U and V , we obtain a new Riemann surface; is this surface the same as the previous

one? Clearly, from a practical view point, both structures should give us the same

results; however, according to the above (informal) definition, these two surfaces

are different since the coverings are not equal. To avoid this type of problems one

defines Riemann surfaces via equivalence classes of coverings satisfying the above

conditions (complex atlases, to be more formal).

Definition. A complex atlas U on a connected surface X is a collection of

pairs U = (Uj , zj)j∈J , where:

(1) Uj are open sets of X;

(2)⋃j∈J Uj = X;

(3) zj : Uj → Wj are homeomorphisms onto open subsets Wj of the complex

plane;

(4) the functions zj satisfy the following compatibility condition: if Uj ∩Uk 6= ∅,then

zk z−1j : zj(Uj ∩ Uk) → zk(Uj ∩ Uk)

is holomorphic.

Observe that the sets zj(Uj ∩ Uk) in (4) are open subsets of C, so one can talk

of zk z−1j being a holomorphic function.

The functions zj are called local coordinates and the pairs (Uj , zj) coordinate

patches. The “transition” functions zkz−1j are known as changes of coordinates.

Definition. Two complex atlases U and V are equivalent or compatible if

their union is a complex atlas. A complex structure on X is an equivalence class

of complex atlases.

1.3.2. Definition. A Riemann surface is a connected surface with a com-

plex structure.


1.3.3. Remarks. 1. Let U be a complex atlas onX and let W denote the atlas

consisting on all local coordinates on X that are compatible with the coordinates of

U . Clearly W contains U , in the sense that any coordinate patch of U belongs to

W. Moreover, W is maximal in the sense that if V is another atlas on X containing

U , then V is contained in W. Clearly U and W are equivalent atlases, so they define

the same complex structure on X. It follows from this observation that to define a

Riemann surface structure on X we just need to give one complex atlas U and then

consider the unique maximal atlas (W in our notation) in the equivalence class of

U .

2. Observe that if U is an atlas on a surface X and (U, z) is a local coordinate of

U , then for any open set V with V ⊂ U , we have that (V, z|V ) is a local coordinate

on X, compatible with U . In other words, restrictions of local coordinates are local

coordinates; we will use this fact frequently.

3. The definition of complex structure generalises to higher dimensions in a natural

way; see §2.4.

4. By a theorem of Rado, Riemann surfaces have a countable basis of open sets.

We will not prove this result here; see, for example [8, pgs. 185-189].

Examples of Riemann surfaces

1.3.4. The complex plane C with the identity function z 7→ z is an example of

a Riemann surface. The maximal atlas for this structure consists of all pairs (U, f),

where U is any open subset of C and f is a one-to-one holomorphic function defined

on U (this is because f is an open mapping, whose inverse is holomorphic). Any open

subset of C with (the complex structure induced by) the restriction of the identity

function is also a Riemann surface. Two important examples of Riemann surfaces

of this type that will appear frequently are the unit disc D = z ∈ C; |z| < 1 and

the upper half plane H = z ∈ C; Im(z) > 0.

1.3.5. A more interesting example is given by the one-point compactification of

the complex plane, C = C∪∞. Recall that this space is constructed by adding one

point to the complex plane, called the point at infinity, ∞. The open sets of C are


the open subsets of C and the sets of the form ∞∪ (C\K), where K is a compact

subset of C (these last sets are the neighbourhoods of ∞). To define a Riemann

surface structure on C we consider two open sets, U1 = C, and U2 = C\0. As

local coordinates we take the homeomorphisms z1 : U1 → C, the identity function

z1(z) = z, and

z2 : U2 −→ C, z2(z) =

1

z, if z 6= ∞,

0, z = ∞.

It is easy to see that the changes of coordinates are holomorphic functions: the

intersection of the two coordinate patches is given by U1 ∩ U2 = C\0, and the

function (z2 z−11 )(z) = 1/z, is holomorphic in U1 ∩ U2, since z 6= 0. This surface is

known as the Riemann sphere. It is not difficult to prove that C is homeomorphic

to the 2-sphere (1.2.1); see exercise 15. We will show later that any Riemann surface

structure on C is equivalent to the one we have just constructed (see §§2.3.5, 2.6.8

and 3.2.8).

1.3.6. Our next example uses a standard technique for constructions of Rie-

mann surfaces, namely quotients (in a broad sense any Riemann surface is a

quotient; see the Uniformization theorem, §3.3.1). Let τ be a complex number

with positive imaginary part and Gτ the group of translations on C of the form

T τn,m(z) = z + n + mτ , where n and m are integers. Denote by Tτ = C/Gτ the

quotient space; that is, Tτ consists of equivalence classes of points of C, where z and

w are equivalent if there exists a transformation T τn,m of Gτ , such that T τn,m(z) = w.

Let π : C → Tτ be the natural quotient map. We put the quotient topology on Tτ :

a set U ⊂ Tτ is open if and only if π−1(U) is open in the complex plane. It is easy to

see that Tτ is a connected surface (exercise 31); moreover, Tτ is a compact surface.

To give a complex structure to Tτ , first observe that there exists a positive number

r, such that d(0, n+mτ) > r, for all pairs of integers (n,m) 6= (0, 0). One can take,

for example, r = 12min1, |τ |. If z0 is a point in C, let U(z0) denote the disc centred

at z0 and radius r. We have that U(z0)∩S(U(z0)) = ∅, for any non-identity element

S of G. Thus the restriction of the quotient mapping π|U(z0) : U(z0) → π(U(z0))

is a homeomorphism (since it is an injective, open mapping), so its inverse gives


us a local coordinate in a neighbourhood of π(z0),(π(U(z0)), π

−1|π(U(z0))

). To

check the compatibility condition, let z0 and z1 be two points in the complex

plane with π(z0) = π(z1). This means that there is an element T τn,m ∈ Gτ , sat-

isfying T τn,m(z0) = z1. We have two local coordinates(π(U(z0)), π

−1|π(U(z0))

), and

(π(U(z1)), π

−1|π(U(z1))

), defined in a neighbourhood of π(z0). It is clear that the

disc U(z0) is mapped to the disc U(z1) by this transformation T τn,m. It follows that

the change of coordinates is given by (π|U(z1))−1 π|U(z0) = T τn,m, which clearly is a

holomorphic function. The atlas U consisting on all pairs of the above form makes

Tτ a Riemann surface, called a torus. One can prove that Tτ is a compact surface

of genus 1. Abel’s theorem (2.9.11) gives us the converse statement: any compact

Riemann surface of genus 1 is of the form Tτ .

Any two tori are homeomorphic (1.2.1), but unlike in the case of the Riemann sphere,

there are many different Riemann surface structures on a torus (that is, not any two

complex structures on a torus are equivalent). (3.3).

1.3.7. In the above example the elements of Gτ (other than the identity) do

not have fixed points in C, and thus we can use the projection π to give a complex

structure on the quotient surface. A similar construction is possible for transfor-

mations with “mild” fixed points, as we show next. Consider the unit disc D as a

Riemann surface with the structure induced by the identity function. Let R : D → D

be the rotation of order 4 given by R(z) = iz, and denote by D4 the quotient space

D/ < R >; that is, two points z and w on the unit disc are identified in D4 if

z = Rn(w), for some n = 0, 1, 2, 3 (R4 is the identity map). It is not difficult to see

that D4 is a manifold, homeomorphic to a disc; D4 is obtained by taking the shaded

sector in figure 6 and identifying the sides l and l′; Let π : D → D4 be the quotient

l

l′

Figure 6. Branched covering.


map; points on D4 are of the form π(z), where z is a point in the unit disc. We can

now get a coordinate on D4 by the expression ψ4(π(z)) = z4. We need to check that

this mapping is well defined: if π(z) = π(w), then z = inw, where n = 0, 1, 2, 3, and

therefore z4 = w4. Since ψ4 is globally defined on D4 the compatibility condition

does not need to be checked. Thus we have that ψ4 induces a Riemann surface

structure on D4.

This examples shows how to give local coordinates to branched coverings (§1.1.22;

the number 4 above is not important in the construction and can be substituted by

any positive integer).

Holomorphic mappings

1.3.8. The definition of a holomorphic function on an open set of the complex

plane (1.1.1) can be easily generalised to the context of Riemann surfaces.

Definition. A continuous mapping f : X → Y between two Riemann surfaces

is called holomorphic if for every pair of local coordinates, (U, z) on X and (V, w)

on Y , with f(U) ∩ V 6= ∅, the function

w f z−1 : z(U ∩ f−1(V )) → w(V )

is holomorphic.

If the target surface is the complex plane, Y = C, we say that f is a holomorphic

function on X.

A meromorphic function is a non-constant holomorphic mapping f : X → C from

a Riemann surface to the Riemann sphere; the points f−1(∞) are called the poles

of f . We will denote by O(X) and M(X) the sets of holomorphic and meromorphic

functions on the Riemann surface X, respectively.

Since the above definition is local some important properties of holomorphic

functions defined on C are satisfied by holomorphic mappings on Riemann surfaces.

For example, the Maximum Modulus Principle, the Open Mapping Theorem (1.1.9)


and the Identity Principle (1.1.6) hold in the setting of Riemann surfaces. Sim-

ilarly, a holomorphic, bijective mapping between two Riemann surfaces, called a

biholomorphism, has a holomorphic inverse (1.1.10).

1.3.9. Holomorphic functions of one variable are holomorphic mappings when

we consider open sets of the complex plane as Riemann surfaces (with the identity

as the coordinate function). One such example is given by a polynomial p : C → C.

We can extend p to the Riemann sphere by defining p(∞) = ∞. To see that this

extended mapping is holomorphic assume that p is given by p(z) = anzn + · · ·+ a0,

where aj are complex numbers, and an satisfying an 6= 0. It is clear that we need

to prove only that the extended function, also denoted by p, is continuous and

holomorphic at the point ∞. Continuity follows from standard results of topology [4,

theorem 11.4, pg. 33]. To show that p is holomorphic at ∞ consider the local

coordinate z2 defined in 1.3.5. The function g = z2 p z−12 = 1/p(1/z) has the

following expression:

g(z) = (z2 p z−12 )(z) =

zn

an + · · ·+ a0zn.

Since an 6= 0 we have that g is holomorphic in a neighbourhood of 0, so p is a

holomorphic mapping on the Riemann sphere. Actually, p is a meromorphic function

whose only pole lies at the point ∞. It also follows from the above expression that

the order of the pole at ∞ is n; we will use this fact in 1.3.14.

The mapping π : C → Tτ defined in 1.3.6 is holomorphic. This is easy to see

since local coordinates on Tτ are given precisely by the local inverses of π. The

quotient mapping π : D → D4 of 1.3.7 is also a holomorphic mapping; its expression

in local coordinates is given by z 7→ z4.

Degree of a holomorphic mapping

1.3.10. The following result is an direct consequence of 1.1.11.

Proposition. Let f : X → Y be a non-constant holomorphic mapping between

two Riemann surfaces. Let p be a point of X, and q = f(p). Then there exist local

coordinates (U, z) and (V, w), on X and Y respectively, such that:


(1) p ∈ U , z(p) = 0; q ∈ V , w(q) = 0;

(2) f(U) ⊂ V ;

(3) the function w f z−1 is of the form ξ 7→ ξn for some integer n (the integer

n depends on f and p, but it is independent of the choices of local coordinates p).

1.3.11. The integer n in the above proposition is called the ramification

number of f at p. The branching number (of f at p), denoted by bp(f), is

defined as bp(f) = n− 1. Observe that if p′ ∈ U , with p′ 6= p, then bp′(f) = 0. Thus

the set of points with positive branching number is discrete; in particular, if X is

a compact surface, it is a finite set. We also have that for any point q′ ∈ V \q,the number of points in f−1(q′) ∩ U is equal to n. If we count multiplicities, as

it is usually done in Complex Analysis, we have that every point in f(U) has n

preimages in U . It follows from this remark that the total number of preimages of

any point q0 of Y is then given by∑

(bp0(f) + 1), where the sum is taken over all

points p0 ∈ X with f(p0) = q0. In the case of compact surfaces this sum is finite,

and independent of the point q0, as the next result shows.

Proposition. Let f : X → Y be a non-constant holomorphic mapping between

compact Riemann surfaces. Then there exists a positive integer d, called the degree

of f , such that any point q ∈ Y has precisely d preimages counted with multiplicity.

In other words, for any q ∈ Y ,

∑

f(p)=q

(bp(f) + 1) = d.

1.3.12. To prove the above proposition we need the following result.

Lemma. Let f : X → Y be a non-constant holomorphic mapping between

two Riemann surfaces. Assume that X is compact. Then Y is compact and f is

surjective.

Proof. Since f is not constant we have that f(X) is open in Y (Open Mapping

Theorem, 1.1.9). On the other hand, since X is compact f(X) is compact in Y ,

and therefore closed. Since Y is connected and f(X) is not empty we have that

f(X) = Y .


1.3.13. Proof of proposition 1.3.11. Consider the decreasing sequence of

subsets of Y given by

Yn = q ∈ Y ;∑

f(p)=q

(bp(f) + 1) ≥ n,

where n is a positive integer. Clearly Yn consists of the points of Y with at least n

preimages, counted with multiplicities. By the previous lemma we have that Y1 = Y

(so at least one of these sets is not empty).

It follows from proposition 1.3.10 that Yn is an open subsets of Y , for all n. We

will show that Yn are closed sets as well. Fix n and let ykk be a sequence of points

in Yn, with limit point y0. Since the number of points of X with positive ramification

number is finite we can assume that f−1(yk) consists of at least n distinct points,

say x1k, . . . , x

nk. By the compactness of X, for each fixed j = 1, . . . , n, there

exists a convergent subsequence of the sequence xjkk. After renaming subindices

if necessary, we get n points, x1, . . . , xn, such that xjk converges to xj (as k → ∞).

Clearly f(xj) = y0 by the continuity of f . If the points xj are distinct we have that

y0 has at least n preimages; that is, y0 belongs to Yn. Assume on the other hand

that some limit points coincide, say x1 = x2; that is, the sequences x1kk and x2

kkconverge to the same limit point x1. Using again 1.3.10 we get that f should be at

least 2−to-1 in a neighbourhood of x1 (see remark after the proof). Thus bx1(f) ≥ 1,

and this point counts as (at least) two preimages of y0. We obtain in this way that

y0 has at least n preimages; that is, y0 ∈ Yn, and therefore Yn is closed.

Since Y is connected, each Yn is either empty or equal to Y . Choose any point

y ∈ Y , and let d be the number of preimages of y. Then we have that y ∈ Yd but

y /∈ Yd+1. It follows that Yd = Y . Since Yn+1 ⊂ Yn, the sets Yn are empty for n > d,

which completes the proof.

Remark. Suppose f : D → D satisfies f(0) = 0. By 1.3.10 we can assume

f(z) = zn. If n = 1, then f is the identity, so there cannot exist two distinct

sequences of points in D, say zkk and wkk, converging to 0 and such that f(zk) =

f(wk). So we have that if such sequences exist then n ≥ 2. Similarly, if there are


three sequences satisfying similar conditions then n ≥ 3. This should clarify the

proof above.

1.3.14. To give a concrete application of the above results we consider a poly-

nomial of degree n, say p(z) = anzn + · · · + a0, where an 6= 0. We have seen that p

can be extended to a holomorphic mapping on the Riemann sphere, whose only pole

of order n lies at the point ∞. It follows that p has degree n; thus p has n zeroes,

counted with multiplicity: this is the Fundamental Theorem of Algebra. We also

see that the degree of a polynomial (in the sense of Algebra) is equal to the degree

when considered as a holomorphic mapping of the Riemann sphere.

The quotient mapping π of 1.3.7 has degree 4. Although the surfaces in this

example are not compact, proposition 1.3.11 applies to a wider class of mappings

(proper mappings); see [8, pg. 29] (exercise 21).

Proposition. Let f : C → C be a non-constant holomorphic mapping. Then

f is a rational function; i.e. there exist two polynomials p and q, with no common

zeroes, such that f(z) = p(z)/q(z). Moreover, the degree of f is the maximum of

the degrees of p and q.

Proof. We leave it as an exercise (exercise 22).

Corollary. The automorphisms (biholormophic self-mappings) of C are the

Mobius transformations; that is, the mappings of the form T (z) =a z + b

c z + d, where

a, b, c, d are complex numbers satisfying ad− bc 6= 0.

Proof. From the above proposition we have that if T is a biholomorphism

of C then it must be a rational function of degree 1. The condition ad − bc 6= 0 is

equivalent to the fact that the numerator and denominator of T do not have common

zeroes.

Triangulations and the Riemann-Hurwitz Formula

1.3.15. In the proof of the next proposition we will assume that compact Rie-

mann surfaces have non-constant meromorphic functions (a non trivial fact that will


be proved in 2.3.4).

Proposition. Compact Riemann surfaces are triangulable.

Proof. Let f : X → C be a non-constant meromorphic function and let S

denote the set of critical points of f ,

S = p ∈ X; bp(f) > 0.

Since X is compact S is a finite set. Let S ′ be the set of critical values, S ′ = f(S).

If T is any triangulation of C we can subdivide it to obtain another triangulation,

say T ′, such that all points of S are vertices of T ′ (see figure 7), and at most one

vertex of any triangle of T ′ lies in S ′. The triangles of T ′ lift via f to triangles in

z0

z0

Figure 7. Subdivision of triangles (z0 is a critical value).

X, i.e. if T is a triangle of T ′ then f−1(T ) is a disjoint union of triangles in X. This

follows from the fact that f : X\f−1(S) → Y \S ′ is a covering map and the interiors

of the triangles are simply connected. Similarly, the (open) edges lift to edges. Thus

we can lift T ′ to a triangulation on X.

1.3.16. If f : X → Y is a holomorphic mapping between compact surfaces of

degree d, for any point q in Y we have that the number of points in the set f−1(q)

(counted without multiplicities) is given by d−∑f(p)=q bp(f).

The total branching number of a holomorphic mapping f : X → C, where

X is a compact surface, is defined by B(f) =∑

p∈X bp(f). Observe that this sum

takes place over only finitely many points.

Proposition (Riemann-Hurwitz relation). Let f : X → Y be a holomorphic

mapping of degree d between compact surfaces of genera g and g′ respectively. Then

the following relation holds:


2g − 2 = d (2g′ − 2) +B(f).

Proof. As in the previous proof we consider the sets S = p ∈ X; bp(f) > 0 of

critical points and S ′ = f(S) ⊂ Y of critical values of f . Let T be a triangulation of

Y such that all points of S ′ are vertices of it, and at most one vertex of any triangle

lies in S ′. Let T denote the triangulation of X obtained by lifting T via f−1. We

have that any triangle of T lifts to d triangles on X, and similarly with the edges. So

if T has F triangles (faces), E edges and V vertices, then T will have d F triangles

and dE edges. The vertices that do not belong to S ′ have d distinct preimages. If

y ∈ S ′, from the observation before the proposition we see that f−1(y) consists of

d−∑f(x)=y bx(f) points. Thus f−1(S ′) consists of dn−B(f) points, where n is the

number of points in S ′. So T has dV −B(f) vertices. Applying the Euler-Poincare

characteristic formula to the triangulations T and T we get F − E + V = 2 − 2g′

and dF − dE + dV − B(f) = 2 − 2g. Putting these two expressions together we

obtain the desired relation.

Corollary. With the hypothesis and notation of the above proposition we have

1. B(f) is an even integer;

2. g ≥ g′.

As an application of this corollary we see that there do not exist any non-constant

holomorphic mappings from the Riemann sphere to a surface of positive genus.

1.4. Differential Forms

The definition of Riemann surface (1.3.2) can be easily modified to obtain other

type of structures on a (topological) manifold. For example, if we require changes

of coordinates to be smooth functions we have the concept of smooth manifolds

[3]. In particular Riemann surfaces are smooth manifolds (1.1.4) and thus we can

apply the results of Differential Geometry to them. In this section we recall some

definitions and theorems about smooth manifolds and obtain some consequences

for the case of Riemann surfaces. We start with a “working” definition of forms,

following [7]; although this approach might look quite formal, it is short and good

1.4. DIFFERENTIAL FORMS 29

enough to state the theorems we need. Nevertheless, in the third subsection we

explain a different point of view, as develop in [8], and its connections with ours as

well as the Differential Geometry definitions. The complex structure enters in the

second part of this section, where we study harmonic and holomorphic forms. The

fourth part explains integration of forms on a surface; two important results, Stokes’

(1.4.22) and Sard’s (1.4.23) theorems, are stated at the end of that part. We finish

this section with meromorphic forms, a natural generalisation of holomorphic forms;

the Residues theorem, a fundamental tool that we will use in the next chapters, is

obtained as an easy consequence of Stokes’ theorem.

Some of the results of Differential Geometry require that manifolds have a count-

able basis of open sets; as remarked in 1.3.2, this is always true for Riemann surfaces.

Differentiable functions and forms

1.4.1. Recall that by a smooth function we mean a function with derivatives

of all orders (class C∞).

Definition. A continuous function f : X → C defined on a Riemann surface

X is said to be smooth (or differentiable) if for any coordinate patch (U, z), the

function (f z−1) : z(U) → C is smooth (as a function defined on an open set of R2

and with values in R2).

We will denote by C(X) and S(X) the sets of continuous and differentiable functions

on X, respectively. Similarly C(U) and S(U) denote the sets of continuous and

differentiable functions on an open subset U of X (see also 1.5).

Using local coordinates (s, t) on the R2 we define two operators on smooth functions,

the partial derivatives, given by the following expressions:

fx(p) =∂f

∂x(p) =

∂(f z−1)

∂s(z(p)), and fy(p) =

∂f

∂y(p) =

∂(f z−1)

∂t(z(p)),

where p ∈ U . From a Complex Analysis point of view it is more natural to consider

the operators fz and fz, defined as follows (see also 1.1.2):

fz =∂f

∂z=

1

2

(∂f

∂x− i

∂f

∂y

), and fz =

∂f

∂z=

1

2i

(∂f

∂x+ i

∂f

∂y

).


Observe that, by the Cauchy-Riemann equations a function is holomorphic if and

only if fz = 0.

1.4.2. The most natural way of defining forms on a surface is by introducing

the tangent plane and then considering its dual. However that would take us into

a course on Differential Geometry, which is outside the scope of this book. As

mentioned in the introduction to this section, we provide a “practical” approach,

and describe later in this section the relationship with other points of view.

A (smooth) 1-form ω on X is an assignment of two smooth complex valued

functions, f and g, to each local coordinate (U, z), such that the following invariance

property is satisfied. If (V, w = x + iy) is another local coordinate on X, with

U ∩ V 6= ∅, and ω is given by f and g on V , then

(1) f = f∂x

∂x+ g

∂y

∂x, g = f

∂x

∂y+ g

∂y

∂y,

on U ∩ V . We will write ω = fdx + gdy for a 1-form. The invariance property is

usually expressed by the equality f dx + g dy = f dx + g dy. The space of 1-forms

on X will be denoted by S1(X); similarly we use S1(U) for the set of 1-forms on an

open subset U of X.

The sets S1(U) are free modules of rank 2 over the ring of C∞ complex valued

functions, with dx, dy being a basis of this module (that any form can be expressed

on U as fdx+ gdy, where f and g are smooth functions). We define two forms on

U by

dz = dx+ idy and dz = dx− idy.

It is easy to check that the pair dz, dz is also a basis of S1(U) (as a vector space

over R). Using dz and dz we can decompose S1(U) in two subspaces in a natural

way: S1(U) = S(1,0)(U) ⊕ S(0,1)(U), where S(1,0)(U) (respectively S(0,1)(U)) is the

subspace generated by dz (respectively dz). Forms in S(1,0)(U) are called of type

(1, 0); in the local coordinate z they are of the form f(z) dz, where f is a smooth

function. Similarly the elements of S(0,1)(U) are called forms of type (0, 1).

The importance of this decomposition lies in the fact that it does not depend on

local coordinates: let (U, z) and (V, w) be a coordinate patches on X with V = U .


Assume that ω = f dz is an element of S(1,0)(U). Let ω = fdw + gdw be the

expression of ω with respect to the coordinate w. We then have

f = f∂z

∂w, g = f

∂z

∂w;

(this is similar to the expression (1) above; we have g = 0). Since the change of

coordinates in holomorphic, we have from Cauchy-Riemann equations that ∂z/∂w =

0, so g = 0. This shows that ω is also an element of S(1,0)(V ).

1.4.3. A (smooth) 2-form α on X is an assignment of a smooth function f to

each local coordinate (U, z), satisfying the following invariance property: with the

above notation, if α is given by the function f on V , then

f = f∂(x, y)

∂(x, y)= f

(∂x

∂x

∂y

∂y− ∂x

∂y

∂y

∂x

).

We write α = fdx ∧ dy or α = fdx dy for a 2-form; we will denote the space of

2-forms by S2(X). We follow the convention dy∧dx = −dx∧dy (dx dy = −dy dx);

see below (1.4.5) for an explanation.

We will frequently use the word form to refer to 1 and 2 forms when it is clear from

the context the type of the form.

Exterior product and exterior derivative

1.4.4. The exterior product of two 1-forms, ωj = fjdx + gjdy, j = 1, 2, is

the 2-form defined by

ω1 ∧ ω2 = (f1g2 − f2g1) dx ∧ dy.

This operation ∧ : S1(X) × S1(X) → S2(X) satisfies (and it is characterised by)

the following properties:

(ω1 + ω2) ∧ ω3 = (ω1 ∧ ω3) + (ω2 ∧ ω3),

(λω1) ∧ ω2 = λ(ω1 ∧ ω2),

ω2 ∧ ω1 = −ω1 ∧ ω2,


for forms ωj, j = 1, 2, 3 and λ ∈ C.

We have that the exterior product of dz and dz is given by

dz ∧ dz = (dx+ idy) ∧ (dx− idy) = −2i dx ∧ dy.

Using this equation, if ωj = ujdz + vjdz, j = 1, 2, are 1-forms, we have

ω1 ∧ ω2 = (u1v2 − u2v1) dz ∧ dz.

1.4.5. The exterior product of forms is a particular case of the following general

algebraic construction. Given a complex vector space V of dimension n let W be

a vector space of dimension d =(n2

), with basis w1, . . . , wd; we define a mapping

∧ : V × V → W by

(vj , vl) 7→wi+j−n if 1 ≤ j < k ≤ n,

(vl, vj) 7→ − wi+j−n if 1 ≤ j < k ≤ n,

(vi, vi) 7→ 0 for i = 1, . . . , n,

and extend it linearly to V × V . Denote the image of (vj, vk) by vj ∧ vk. The space

W is usually denoted by V∧V or

∧2 V , and is called the 2nd exterior product of

V .

For a (fixed) point x of X we denote by S1x(X) the vector space generated by dx

and dy. Similarly we denote by S2x(X) the vector space generated by dx∧dy. Then

we have that S2x(X) = S1

x(X)∧S1x(X).

Since S1 is a 2 dimensional space the space S2 has dimension 1.

A similar definition for the mth exterior product is easy to obtain from this con-

struction. The dimension of∧m V is

(nm

)for m ≤ n, and 0 if m > n; hence there

are no 3-forms on a surface. In general, one can define forms of order at most n on

an n dimensional manifold. See [12, pg. 731] and [11, pg. 391] for an Algebraic

approach, or [25] for a Differential Geometric point of view.

1.4.6. The exterior derivative d, is an operator on functions and forms de-

fined in the following way. For f ∈ S(X) we set

df = fxdx+ fydy;


for a 1-form ω = fdx+ gdy we define

dω = fydy ∧ dx+ gxdx ∧ dy = (gx − fy)dx ∧ dy;

and for α ∈ S2(X) we put dα = 0. The form df is called the differential of the

mapping f . In the basis dz, dz the exterior derivative is given by:

df = fzdz + fzdz, dω = (vz − uz) dz ∧ dz,

where ω = udz + vdz.

One can easily check that df = 0 if and only if f is constant.

We can decompose the action of d on functions in its (1, 0) and (0, 1) parts,

∂ : S(X) →S(1,0)(X) ∂ : S(X) →S(0,1)(X)

f 7→fzdz f 7→fzdz.

It is clear that a function is holomorphic if and only if ∂f = 0. Since d2 = dd = 0,

we have ∂2 = ∂2 = ∂∂+ ∂∂ = 0 (observe that the operators ∂ and ∂ anti-commute).

1.4.7. The conjugation operator is an operator on the space of 1-forms, ∗ :

S1(X) → S1(X) defined by:

∗ (fdx+ gdy) = gdx− fdy,

or in complex notation,

∗ (udz + vdz) = −iudz + ivdz.

Observe that ∗(∗ω) = −ω.

1.4.8. A 1-form ω is called closed if dω = 0, and exact if ω = df for some

smooth function f . We say that ω is co-closed (co-exact) if ∗ω is closed (exact,

respectively).

Since d2 = 0 exact forms are closed. The next lemma is an important result that

shows that closed forms are locally exact (see the next paragraph for the meaning

of the word locally in this context). We will prove it in 1.5.24.

Lemma (Poincare Lemma). Let D ⊂ C be an open disc, and ω a closed 1-form

on D. Then ω is exact.


Holomorphic and Harmonic Forms

1.4.9. A 1-form ω is called holomorphic if it is given locally by ω = df ,

where f is a holomorphic function. Locally means that for every point p of X there

exists a neighbourhood V , and a holomorphic function f : V → C, such that ω = df

on V . The function f and the set V might depend on the point p. The space of

holomorphic forms is denoted by Ω(X). In the next chapter we will show that if X

is a compact Riemann surface of genus g then Ω(X) has (complex) dimension g.

1.4.10. Proposition. A form ω is holomorphic if and only if it can be written

as ω = u dz, where u is a holomorphic function.

Proof. Since f is holomorphic we have fz = 0, and therefore ω = df = fz dz.

To prove the converse statement consider a local coordinate (V, z) near a point p, and

let ω be given by ω = u dz, where u is a holomorphic function V . By shrinking V if

necessary we can assume that z(V ) is a disc on C. The function u z−1 : z(V ) → C

is holomorphic and therefore it has a holomorphic primitive [20, chapter 10]; that

is, there exists a holomorphic function on z(V ), say g, such that g′ = u z−1. The

functionf = g z is holomorphic on V (f z−1 = g) and satisfies ω = df .

1.4.11. The Laplacian operator ∆ : S(X) → S2(X) assigns a 2-form to a

smooth function f by the rule ∆f = (fxx + fyy) dx ∧ dy. It is easy to see that

∆ = d ∗ d = 2 i ∂ ∂, where ∗ is the conjugation operator defined in 1.4.7. A function

f is called harmonic if ∆f = 0. Harmonicity is a property that depends on the

Riemann surface structure of X; thus it might happen that a function defined on

a surface is harmonic for certain Riemann surface structure but not for a different

structure (see exercise 66 for an example). A 1-form ω is called harmonic if it can

be written locally as ω = df , where f is a harmonic function.

1.4.12. Proposition. A 1-form is harmonic if and only if it is closed and

co-closed.

Proof. If ω = df is harmonic then

dω = d2f = 0


and

d(∗ω) = d ∗ df = ∆f = 0,

so ω is closed and co-closed.

Conversely, if ω is closed then ω is locally exact (Poincare lemma, 1.4.8); that

is, ω = df for a smooth function f . If ω is co-closed we get d ∗ ω = ∆f = 0, so the

function f is harmonic.

1.4.13. Proposition. A 1-form ω = udz + vdz is harmonic if and only if u

and v are holomorphic functions.

Proof. We have

dω = (vz − uz)dz ∧ dz

and

d(∗ω) = d (−iudz + ivdz) = i (vz + uz) dz ∧ dz.

So ω is closed and co-closed (therefore harmonic, by the previous proposition) if

and only if uz = vz = 0. As we have remarked earlier, uz = 0 is equivalent to u

being holomorphic. On the other hand, vz = (v)z, so we vz = 0 if and only if v is

holomorphic.

1.4.14. If f is a holomorphic function defined on the complex plane then its

real and imaginary parts (1.1.1) are harmonic functions. The next proposition is

the equivalent result for holomorphic 1-forms.

Proposition. A form ω is holomorphic if and only if it can be written as

ω = τ + i (∗τ), for some harmonic form τ .

Proof. We have seen (1.4.10) that if ω is holomorphic then it can be written

(locally) as ω = u dz, for some holomorphic function u. Let τ be defined by τ =

12(u dz − u dz). By the previous result τ is harmonic, and we have

i (∗ τ) =i

2(−i u dz + i u dz) =

1

2(u dz + −u dz) ,

so τ + i (∗ τ) = ω.


Conversely, if τ is harmonic we can write τ = ω1 +ω2, where ω1 and ω2 are holo-

morphic forms. Then i(∗τ) = i(−iω1 +iω2) = ω1−ω2; thus τ +i(∗τ) is holomorphic.

A different approach to forms

1.4.15. In this subsection we explain the definition of forms as given in [8] and

its relation with the definitions in the previous subsections.

Given a point p of X, consider the collection of pairs (U, f), where U is a neigh-

bourhood of p and f : U → C is a smooth function. Equivalently, take the union⋃S(U), as U varies over the neighbourhoods of p. Define an equivalence relation

∼p in this union by identifying (U, f) and (V, g) if there exists a neighbourhood W

of p, with W ⊂ U ∩ V and such that f |W = g|W . Let Sp =(⋃

p∈U S(U))/ ∼p

denote the set of equivalence classes (see 1.5.15 for a general construction of this

type). Consider now two subspaces of Sp: Sp(1), consisting of all equivalence classes

of functions that vanish at p, and Sp(2) containing the equivalence classes of func-

tions satisfying f(p) = fx(p) = fy(p) = 0. The quotient space T ∗pX = Sp(1)/Sp(2)

is called the cotangent space of X at p. The union of all these spaces as p varies

over X is the cotangent bundle of X, T ∗X =⋃p∈X T

∗pX. There is a natural pro-

jection π : T ∗X → X given by π(T ∗pX) = x (in the language of §2.4, the cotangent

space is a smooth vector bundle over X of rank 2). If f is a function defined in a

neighbourhood of p the class of f − f(p) in T ∗pX is called the differential of f at

p, written as dpf . For a local coordinate z = x + iy at p, its real and imaginary

parts are functions defined near p; let dpx and dpy be the corresponding elements

(differentials) of T ∗pX.

Claim. The differentials dpx, dpy form a basis of T ∗pX.

If dpf is an arbitrary element of this vector space we know from basic Calculus

that there exists a function g satisfying f = a(x − x(p)) + b(y − y(p)) + g, and

g ∈ Sp(2). Thus dpf = a dpx+ b dpy. We see from this formula that a = ∂f∂x

(p) and

b = ∂f∂y

(p). To prove the linear independence assume that r dpx + s dpy = 0. Then

r(x− x(p)) + s(y − y(p)) belongs to Sp(2). Taking partial derivatives evaluated at

p, we see that r = s = 0.


An important result obtained in the proof of the above claim is the following ex-

pression for the differential of a function:

dpf =∂f

∂x(p) dpx+

∂f

∂y(p) dpy.

1.4.16. A 1-form ω is a mapping fromX to its cotangent space, ω : X → T ∗X,

such that ω π = IdX . This means that ω(p) belongs to the cotangent space of

X at p. So we can write ω = f dx + g dy i.e. ω(p) = f(p) dpx + g(p) dpy. We

say that ω is smooth if f and g are smooth functions. The wedge product of two

1-forms is defined as in 1.4.4. By (T ∗p )2X we mean T ∗

pX∧T ∗pX, the space consisting

of all exterior products of 1-forms at p. Since dpx ∧ dpy = − dpy ∧ dpx, the space

(T ∗p )2X has dimension 1. A 2 -form is a mapping α : X → (T ∗)2X, such that

α(p) ∈ (T ∗p )2X. We have that α is locally given by α = f dx ∧ dy. If f is smooth

we say that α is a smooth 2-form.

Clearly the above approach agrees with our original definitions, at least locally.

From the usual Calculus formulæ of the derivative of the composition of two func-

tions one can get the invariance property (behaviour under changes of coordinates)

of 1 and 2-forms.

1.4.17. The relation between this definition and the usual one given in Diffe-

rential Geometry is as follows: the set Sp has the structure of a vector space of

dimension 2 over R. Let TpX be the set of linear mappings L : Sp → R, satisfying

the Leibnitz rule L(f g)(p) = (Lf)(p) g(p) + f(p) (Lg)(p); TpX is called the tangent

space of X at p. The cotangent space T ∗pX is simply the dual of TpX (from where

the notation comes).

Integration of forms

1.4.18. The invariance property of forms, which resembles the change of vari-

ables used in Calculus, allows us to define integrals of forms on surfaces, as we show

in this subsection.


A function f : [a, b] → C is said to have right derivative at a point t if the

limit

(f ′)+(t) = lim

h→0+

f(t+ h) − f(t)

h

exists. Similarly one defines the left derivative.

Definition. A curve γ : [a, b] → H (continuous mapping) is called piecewise

smooth if there exist points a = t0 < t1 < · · · < tn = b, such that γ is smooth on

each of the intervals (tj, tj+1) and it has right and left derivatives at the points tj ,

whenever it is possible to compute them (for example, at a we can talk only of the

right derivative).

If c : [0, 1] → X is a curve on a Riemann surface X, we say that c is piecewise

smooth if we can cover its image c([0, 1]) with coordinate patches (Uj , zj)j∈J , such

that zj c is piecewise smooth (for each j).

For the rest of this section we will assume that all curves are piecewise smooth,

unless otherwise stated.

1.4.19. Subdividing [0, 1] into smaller intervals if needed we can assume that

for each subinterval [tj , jj+1] there exists a local coordinate patch (Uj , zj = xj +iyj),

such that c([tj, tj+1]) ⊂ Uj . Let ω be a 1-form on X, given by ω = fjdx + gjdy on

Uj . We define

∫

c

ω =

n∑

j=1

∫ tj+1

tj

(fj(c(t))

∂(xj c)∂t

(t)gj(c(t))∂(yj c)

∂t(t)

)dt.

We leave to the reader to check that this definition is independent of the choices of

local coordinates or subdivision of the interval [0, 1].

1.4.20. See [25] for proofs of the next two results.

Proposition. Let ω be a closed 1-form on X, and cj : [0, 1] → X, j = 1, 2,

two paths with the same end points. If c1 is homologous to c2 (the cycle c1 − c2 is a

boundary) then∫

c1

ω =

∫

c2

ω.


Corollary. If ω be closed 1-form, and c1 and c2 two homotopic paths on X

(with the same end points), then∫c1ω =

∫c2ω.

1.4.21. The integration of a 2-form requires a little more work. First of all,

observe that one cannot speak of the value of a form at a point. However, since

changes of coordinates have non-zero Jacobians, it makes sense to say whether a

form vanishes or not at a given point. More precisely, if α ∈ S2(X), we define its

support by

supp(α) = p ∈ X; α(p) = 0.

We have that supp(α) is a well defined set. Assume that for a certain form α the set

supp(α) is compact and contained in a single coordinate patch, say (U, z = x+ iy).

We define the integral of α by the expression

(2)

∫

X

α =

∫

z(U)

(f z−1) ds dt,

where (s, t) are coordinates on R2. As in the case of 1-forms, the invariance property

can be used to prove that this integral is well defined (its value does not depend on

the local coordinate z). More generally, if α is a 2-form with compact support we

cover supp(α) by finitely many coordinates patches, say (Uj , zj), j = 1, . . . , n. Let

φ1, . . . , φn be a partition of unity subordinate to this covering (1.1.23). We then

set

(3)

∫

X

α =

n∑

j=1

∫

X

fj α,

where α is given by fj dxj ∧ dyj on Uj . Observe that the forms φj α have compact

support contained in Uj ; their integrals are defined by the expression (2). One can

check that (3) is independent of the choices made to compute it.

1.4.22. One of the most important results on integration of forms is given by

the following theorem.

Theorem (Stokes). Let ω be a 1-form on a Riemann surface X. Let U be an

open subset of X with compact closure and smooth boundary ∂U (that is, ∂U consists

of a finite collection of smooth curves). Then∫

U

dω =

∫

∂U

ω.


Corollary. Let X be a compact Riemann surface, and ω ∈ S1(X). Then∫X

dω = 0.

The integration on the boundary of U has to be done with certain orientation.

We will use Stokes’ theorem in two types of settings: in one case we consider two

disjoint domains U1 and U2, with a common boundary, given by a smooth curve c,

∂U1 = ∂U2 = c. Then∫∂U1

ω = −∫∂U2

ω. In the other situation (2.2.1) the domain

of integration will be a disc, U = z ∈ C; |z| < R with the standard orientation of

C; its boundary, the circle of centre 0 and radius R, is oriented clockwise (opposite to

the standard orientation). For more details on Stokes’ theorem and its applications

see [25].

1.4.23. We will need two more theorems of Differential Geometry. The first

one tells us that the set of “bad” (critical) values of a smooth map is small in

measure. More precisely, let f : X → R be a smooth proper map defined on a

(smooth) surface X. Recall that f is called proper if f−1(K) is compact for any

K ⊂ R compact. Let p be a point of X, and z = x + iy a local coordinate near p.

We say that p is a critical point of f is fx(p) = fy(p) = 0. Critical points are well

defined because changes of coordinates have non-vanishing derivatives. A critical

value is the image of a critical point under f .

Theorem (Sard). Let f : X → R be a smooth proper mapping defined on a

smooth surface. Let C be the set of critical values of f . Then C has zero measure

in R.

1.4.24. The next result classifies all smooth curves (1-dimensional mani-

folds) [15].

Theorem. Any compact, smooth 1-dimensional manifold is diffeomorphic to a

circle.

Meromorphic forms

1.4.25. A 1-form ω is called meromorphic if it can be written locally as

ω = f(z) dz, where f is a meromorphic function. The set of poles of the functions


f are also known as the poles of ω. They are well defined (do not depend on the

function f chosen to represent the form) and form a discrete set of points of X. The

set of meromorphic forms on X will be denoted by M1(X). In a local coordinate

(U, z) the function f has a power series expansion of the form

f(p) =∞∑

j=N

aj (z(p))j , p ∈ U,

which we simplify as f =∑∞

j=N aj zj . The coefficient a−1 is called the residue of

the form ω at the point p, and we will denote it by resp(ω).

Proposition. The residue of a meromorphic form is well defined.

Proof. We need to show that a−1 is independent of f and the coordinate z.

Assume then that z(p) = 0 as above. Let r and ǫ be positive numbers such that

D(0, r+ ǫ), the disc of centre 0 and radius r+ ǫ is contained in z(U). Since the poles

of ω form a discrete set we can further assume that the only singularity of the form

ω in z−1(D(0, r + ǫ)) = q ∈ U ; |z(q)| < r + ǫ lies at the point p. Consider now

the smooth curve c : [0, 1] → U given by c(θ) = z−1(reiθ). One easily sees that

a−1 =1

2πi

∫

c

ω.

Since this formula gives an way of computing a−1 in terms of the integral of ω over

a path we see that the residue does not depend of local coordinates (since path

integrals are coordinate-independent).

1.4.26. The following result is a consequence of Stokes’ Theorem.

Theorem (Residues Theorem). Let ω be a meromorphic form on a compact

surface X. Then∑

p∈Xresp(ω) = 0.

Proof. First of all, since X is compact and the set of poles of ω is discrete we

have that the above sum is finite. So let us assume that these poles are at points

p1, . . . , pn. It is possible to find a simply connected set with smooth boundary, say

U , containing all the poles, as follows: let c′ be a smooth curve passing though the

poles (once by each pole) and take a small, smooth neighbourhood of c′, (which

can be done in local coordinates), as in figure 8. Let c denote the boundary of U .


* * *

D

cc′

Figure 8. Proof of theorem 1.4.26.

We have that c is homotopic to a collection of small “circles” around the poles pj,

similar to the curves used in the proof of the previous result. Thus

∫

c

ω =

n∑

j=1

respjω.

On the other hand the form ω has no poles onX\U so it is holomorphic and therefore

closed. Applying Stokes’ Theorem (1.4.22) we get

0 =

∫

X\Udω =

∫

c

ω,

which completes the proof.

1.5. Sheaf Cohomology

The purpose of this section is to make the reader familiar with the language (and

basic techniques) of sheaf theory that will be used in the statements and proofs of

the results of the next chapter.

Sheaves of abelian groups

1.5.1. Definition. A presheaf of abelian groups F on a Riemann surface

(or topological space) X is an assignment of an abelian group, F(U), to each open

subset U of X, and a collection of group homomorphisms ρUV : F(U) → F(V ), for

every pair of open sets V and U with V ⊂ U , satisfying the following two conditions:

(1) ρUU is the identity homomorphism;

(2) if W ⊂ V ⊂ U are open sets then ρVW ρUV = ρUW .

The mappings ρUV are called the restriction homomorphisms (of the presheaf).

If f ∈ F(U) and V ⊂ U we will write f |V for ρUV (f), or simply consider f as an

1.5. SHEAF COHOMOLOGY 43

element of F(V ) if it is clear from the context. If the sets F(U) have extra structures,

for example they are vector spaces, one can define presheaves of vector spaces, by

requiring the restriction homomorphisms to be linear mappings. Similarly one has

presheaves of rings, fields and other structures. In this book we will work only with

presheaves of abelian groups and vector spaces.

An example of presheaf is given by C, the set of continuous functions, with the

usual restriction of domains of functions: C(U) denotes the space of continuous

functions defined on the open set U . Here we are assuming that the functions take

values in C but one can consider the presheaf of real valued continuous functions

as well. Another examples of presheaves are given by S, O and M, consisting of

smooth, holomorphic and meromorphic functions respectively.

1.5.2. Definition. A presheaf F is called a sheaf if for every open set U of

X, and every collection of open sets Ujj∈J , such that⋃j Uj = U , the following

conditions are satisfied:

(1) if f and g are elements of F(U) such that f |Uj= g|Uj

, for all j ∈ J , then

f = g;

(2) if fj ∈ F(Uj), j ∈ J , and fj |Uj∩Uk= fk|Uj∩Uk

, then there exists an element

f ∈ U with f |Uj= fj .

The examples of presheaves given above are actually sheaves (see exercise 36 for an

example of a presheaf that is not a sheaf). Another example of a sheaf is given by

O∗, the set of nowhere vanishing holomorphic functions

O∗(U) = f : U → C\0; f is holomorphic.

Similarly we can define M∗, the sheaf of “invertible meromorphic functions”; that

is, M∗(U) consists of the meromorphic functions on U which are not identically 0

on any component of U .

For the rest of this section F , G and H will denote sheaves of abelian groups.

Cohomology groups


1.5.3. Let U = Ujj∈J be an open covering of a Riemann surface X. The

nth cochain group, n a non-negative integer, Cn(U ,F), is defined by

Cn(U ,F) =∏

(j0,...,jn)∈Jn+1

F(Uj0 ∩ · · · ∩ Ujn).

Elements of C0(U ,F) are of the form (fj)j∈J , where fj ∈ F(Uj); we will simplify this

notation by writing (fj). Similarly, (fjk) and (fjkl) will denote elements of C1(U ,F)

and C2(U ,F), respectively. The elements of Cn(U ,F) are called n-cochains.

1.5.4. The coboundary operators, δ : Cn(U ,F) → Cn+1(U ,F), for n = 0, 1,

are defined in the following way:

1. if (fj) ∈ C0(U ,F) then δ((fj)) = (gkj), where gjk is given by gjk = fk − fj, in

Uj ∩ Uk; (strictly speaking we mean gjk = fk|Uj∩Uk− fj |Uj∩Uk

; but see above for the

notation);

2. for (fjk) ∈ C1(U ,F) we set δ((fjk)) = (gjkl), with gjkl = fkl − fjl + fjk in

Uj ∩ Uk ∩ Ul (similarly gjkl = fkl|Uj∩Uk∩Ul− fjl|Uj∩Uk∩UL

+ fjk|Uj∩Uk∩Ul).

The operator δ satisfies δ2 = δ δ = 0. To check this, let (fj) be a 0-cochain. Then

we have

δ(δ((fj))) = δ((fk − fj)) = (fl − fk) − (fl − fj) + (fk − fj) = 0.

1.5.5. The groups of cocycles and coboundaries, Z1(U ,F) and B1(U ,F),

are defined respectively as the kernel of δ : C1(U ,F) → C2(U ,F) and the image of

δ : C0(U ,F) → C1(U ,F). A 1-cochain (fjk) is a cocycle if it satisfies fjl = fjk + fkl

in F(Uj ∩ Uk ∩ Ul) (we can “add” the middle subindex). In particular a cocycle

satisfies fjj = 0 and fjk = −fkj . A 1-cochain is a coboundary, or it splits, if

fjk = fk−fj for some 0-cochain (fj). Observe that this is equivalent to the equality

fjk = gj − gk where (gk) is the cocycle given by gk = −fk. The group B1(U ,F) is

a subgroup of Z1(U ,F) because δ2 = 0. Since cochain groups are abelian, it is a

normal subgroup, so the quotient is a group, called the 1st cohomology group of

X (with coefficients in F with respect to the covering U),

H1(U ,F) = Z1(U ,F)/B1(U ,F).


1.5.6. Our goal is to define a group that does not depend of the covering U . To

do this we first introduce an order in the collection of all open coverings of X: the

covering V = Vαα∈A is finer than the covering U = Ujj∈J , written as V < U ,

if for every α ∈ A there exists a j ∈ J , such that Vα ⊂ Uj. Let φ : A → J be a

mapping such that Vα ⊂ Uφ(α). Observe that φ might not be uniquely defined, but

we will show that our constructions are independent of the choice of this refinement

mapping (1.5.7). Since Vα is an open subset of Uφ(α) the mapping φ composed with

the restriction homomorphisms induces a mapping between cochain groups in the

following way:

φUV : C1(U ,F) → C1(V,F)

(fjk) 7→ (fφ(a)φ(b)|Va∩vb).

It is clear that φUV preserves cocycles and coboundaries and thus it induces a mapping

between cohomology groups, which we will denote by φUV as well:

φUV : H1(U ,F) → H1(V,F).

We next show that this mapping, at the cohomology level, does not depend of

the refinement map φ.

1.5.7. Lemma. The mapping φUV : H1(U ,F) → H1(V,F) is independent of

φ.

Proof. Assume ψ : A → J is another refinement map; that is, Vα ⊂ Uψ(α), for

all α ∈ A. Define two mappings by the expressions

K1 : C1(U ,F) →C0(V,F)

(fjk) 7→(fφ(α)ψ(α)|Vα)

and

K2 : C2(U ,F) →C1(V,F)

(fjkl) 7→((fφ(α)ψ(α)ψ(β) − fφ(α)φ(β)ψ(β))|Vα∪Vβ

).


We claim that K2δ − δK1 = ψUV − φU

V . This is a simple computational exercise. Let

(fjk) ∈ C1(U ,F); then

K2(δ(fjk)) =K2(fjk − fjl + fkl) =

=fφ(α)ψ(α) − fφ(α)ψ(β) + fψ(α)ψ(β) − fφ(α)φ(β) + fφ(α)ψ(β) − fφ(β)ψ(β),

and

δ(K1(fjk)) = δ(fφ(α)ψ(α)) = fφ(α)ψ(α) − fφ(β)ψ(β).

If ξ ∈ Z1(U ,F) is a cocycle then K1(ξ) is a 0-cochain with respect to the covering

V, so δ(K1(ξ)) ∈ B1(V,F). On the other hand, if we write ξ = (fjk) we have

K2(δ(ξ)) = fφ(α)ψ(α) + fφ(α)ψ(β) − fφ(α)φ(β) − fφ(β)ψ(β) = fφ(α)ψ(β) − fφ(α)ψ(β) = 0.

Thus (φUV − ψU

V )(ξ) ∈ B1(V,F). But this implies that φUV and ψU

V induce the same

mappings in cohomology.

1.5.8. Lemma. The mapping φUV : H1(U ,F) → H1(V,F) is injective.

Proof. Assume that (fjk) is a cocycle (with respect to U) such that φUV((fjk))

splits. This means that there exists a cochain (hα), satisfying fφ(α)φ(β) = hα−hβ on

Vα ∩ Vβ. Define fj ∈ F(Uj) by fj = fjφ(α) + hα on F(Uj ∩ Vα). For β ∈ A, on the

intersection Uj ∩ Vα ∩ Vβ we have

(hα + fjφ(α)) − (hβ + fjφ(β)) = hα − hβ + fjφ(α) + fφ(β)j = fφ(α)φ(β) + fφ(β)φ(α) = 0,

because (fjk) is a cocycle. Since the sets Uj∩Vα cover Uj as α varies over A, we have

that fj is an element of F(Uj), by property (2) in definition 1.5.2. On Uj ∩Uk ∩ Vαwe get

fj − fk = (fjt(α) + hα) − (fkt(α) − hα) = fjt(α) + ft(α)k = fjk,

and therefore, by property (1) of sheaves, we have that fj − fk = fjk on Uj ∩ Uk;

that is, (fjk) splits.

1.5.9. The mappings φUV can be used to define a cohomology group independent

of the coverings. First of all, if we have three coverings, U , V and W satisfying

W < V < U , it is easy to see that φVW φU

V = φUW . Consider now the disjoint union

of cohomology groups⊔H1(U ,F), where U varies over all possible open coverings

of X. We define an equivalence relation ∼ in this union by identifying ξ ∈ H1(U ,F)


and η ∈ H1(V,F), ξ ∼ η, if φUW(ξ) = φV

W(η) in H1(W,F), for some covering Wfiner than both U and V. The quotient set

H1(X,F) :=⊔

UH1(U ,F)/ ∼,

is called the 1st cohomology group of X (with coefficients in the sheaf F). The

group structure on this set is defined by passing to finer coverings; if µ and ν are two

cohomology classes, choose ξ ∈ H1(U ,F) and η ∈ H1(V,F), such that µ = [ξ] and

ν = [η] (where square brackets denote equivalence classes in H1(X,F)). Then we

define µ+ν = [tUW(ξ)+ tVW(η)], where W is a covering satisfying W < V and W < U .

Observe that such a covering always exists: if U = Ujj∈J and V = Vaa∈A, then

W = Uj ∩ Va(j,a)∈J×A

is finer than U and V. We leave as an exercise to show that this addition in H1(X,F)

is well defined. From the definition of cohomology groups and lemma 1.5.8 it follows

that for any covering U the natural mappings H1(U ,F) → H1(X,F) are injective.

1.5.10. The following result is obvious from these observations, but we state

it as a separate lemma because we will use it frequently.

Lemma. For a sheaf F of abelian groups on a Riemann surface X, the cohomol-

ogy group H1(X,F) vanishes if and only if for any open covering U of X the group

H1(U ,F) vanishes.

1.5.11. Remark. Some authors consider cochain groups as in 1.5.3, but with

the condition that jk 6= jl if k 6= l. For example, if U consists of two open sets, say

U1 and U2, and f is a 1-cochain in their sense, then f consists of f12 and f21, while

with our definitions f will be given by f11, f12, f21 and f22. But it can be proven

that both constructions give the same cohomology groups (if f is a cocycle then it

is determined by f12 and f21 since f11 and f22 must vanish), which are the objects

of our interest.

1.5.12. We have enough material to give an example of a cohomology group.

Proposition. For a Riemann surface X we have H1(X,S) = 0, where S is the

sheaf of smooth functions on X.


Proof. Let U = Ujj∈J be an open covering of X and (fjk) an element of

H1(U ,S). Let φjj∈J be a partition of unity subordinate to U . The functions

fjkφk belong to S(Uj) (figure 9). Define

hj =∑

k

fjkφk.

Then we have

hj − hk =∑

l

fjlφl −∑

l

fklφl =∑

l

(fjl − fkl)φl =∑

l

(fjl + flk)φl =∑

l

fjkφl

=fjk∑

l

φl = fjk.

Observe that these sums are finite (at every point).

UkUj

supp (φj )

Figure 9. Proposition 1.5.12.

The above result also holds for the sheaves of 1 and 2 forms, S1 and S2, as well as

for the sheaves of forms of type (1, 0) and (0, 1), S(1,0) and S(0,1).

1.5.13. The following result is a useful tool to compute cohomology groups.

Theorem (Leray). Let U = Ujj∈J be an open covering of a Riemann surface

X and F a sheaf of abelian groups on X. Assume that H1(Uj ,F) = 0 for all j ∈ J ;

then H1(X,F) ∼= H1(U ,F).

Proof. Observe that the groups H1(Uj,F) are the cohomology groups of the

sets Uj considered as Riemann surfaces. Let V = Vaa∈A be a covering finer than

U and φ : A → J a refinement mapping. We have proved that φUV : H1(U ,F) →

H1(V,F) is injective; we will show that it is surjective as well, and that will prove

the theorem. So given (fab) in C1(V,F), we need to find a cocycle (Fjk) ∈ C1(U ,F),

such that (Ft(a)t(b) − fab) splits (with respect to the covering V).

For a fixed j ∈ J the collection Uj ∩ V = Uj ∩ Vaa∈A is an open covering of

Uj . Since H1(Uj,F) = 0, there exists a cochain (gja) ∈ C0(Uj ∩ V,F), such that


fab = gja − gjb (on Uj ∩ Va ∩ Vb). Set Fjk = gka − gja, on Uj ∩ Uk ∩ Va; by property (2)

in the definition of sheaf (1.5.2), (Fjk) ∈ F(Uj ∩Uk). To check that (Fjk) is actually

a cocycle observe that

Fjk + Fkl = (gka − gja) + (gla − gka) = gla − gja = Fjl,

and then use property (1) in (1.5.2). We claim that (Ft(a)t(b) − fab) splits. Write

ha = gt(a)a . Then we have

Ft(a)t(b) − fab = gt(b)a − gt(a)a − gt(b)a + gt(b)b = g

t(b)b − gt(a)a = hb − ha,

which shows that Ft(a)t(b) − fab splits.

A covering satisfying the hypothesis of the theorem is called a Leray covering.

1.5.14. Consider the subgroup of the group C0(U ,F) given by

Z0(U ,F) = ker(C0(U ,F)

δ→ C1(U ,F)).

If a cochain (fj) satisfies δ((fj)) = 0 then we have fj = fk on Uj ∩ Uk. From the

definition of a sheaf we have that there exists an element f ∈ F(X), such that

f |Uj= fj . Thus Z0(U ,F) can be identified with F(X), for any covering U . The

B0(U ,F) is the trivial group (it does not make sense to talk of 0-coboundaries);

thus we define the 0th cohomology group of X (with coefficients in F) by

H0(X,F) = F(X).

Sheaf associated to a presheaf

1.5.15. We now introduce a standard way of associating a sheaf to a given

presheaf. The reader familiar with continuation of analytic functions in the complex

plane will find similarities between that construction and the one that follows. See

also 1.4.15. Let F be a presheaf of abelian groups on X. For a point p ∈ X, take

the disjoint union⊔U F(U), where U varies over the open neighbourhoods of p.

We define an equivalence relation ∼p

on this union as follows: let f ∈ F(U) and

g ∈ G(V ), then f ∼pg if there exists a neighbourhood W of p, with W ⊂ U ∩ V ,


such that f |W = g|W . Let Fp denote the quotient set, Fp = (⊔UF(U)) / ∼p, called

the stalk of F at p. For f ∈ F(U) we will denote its class in Fp by fp. Let

N(U, p) = fq; q ∈ U, and declare these sets, as U varies over all neighbourhoods

of p, a fundamental set of neighbourhoods of fp in |F| = ⊔q∈XFq. This construction

defines a topology in |F|. The mapping π : |F| → X, defined by π(fp) = f(p) is

continuous (observe that f(p) is well defined, since all elements of Fp agree on p).

For A ⊂ X open, we set |F|(A) = s : A → |F|; s continuous and π s = IdA.This makes |F| a sheaf on X.

Sheaf homomorphisms and sequences

1.5.16. Definition. Let F and G be sheaves of abelian groups on a Riemann

surface X. A sheaf homomorphism α : F → G is a collection of group homomor-

phisms, αU : F(U) → G(U), for each open set U of X, such that if V ⊂ U , then

diagram

F(U)αu

//

ρUV

G(U)

µUV

F(V )αV

// F(V )

commutes. Here ρ and µ denote the restriction homomorphisms of the sheaves Fand G respectively.

If it is clear from the context, we will drop the subindex in αU and simply write

α : F(U) → G(U).

If F and G are sheaves of some other structure then the mappings αU are required

to be compatible with that structure; for example, for sheaves of vector spaces the

mappings αU should be linear mappings.

1.5.17. An example of a sheaf homomorphism is provided by the exterior de-

rivative, d : S → S1, f 7→ df . The ∂ and ∂ operators are also sheaf homomorphisms.

The kernel of ∂ consists of the holomorphic functions (on U) and therefore it is a

sheaf. In general, if we define

K(U) = ker(αU : F(U) → G(U)

),


we have that K is a sheaf, called the kernel sheaf.

1.5.18. Another example of sheaf homomorphism that we will use in the next

chapter is given by the exponential map, exp : O(U) → O∗(U), exp(f) = e2πif . The

kernel of this sheaf consists of the locally constant functions with integer values,

which we denote by Z as well: a continuous function f : U → C belongs to Z(U) if

it takes integer values and every point of U has a neighbourhood where f is constant.

This is equivalent to require that f is constant on each connected component of U .

In particular, since Riemann surfaces are connected, Z(X) consists of the constant

functions with integer values, which is clearly isomorphic to Z. See also exercise 37.

1.5.19. Definition. Given two sheaf homomorphisms α : F → G and β :

G → H, we say that the sequence F α→ G β→ H is exact if the following two

conditions are satisfied (for every open set U of X):

(1) βU αU = 0;

(2) if g ∈ G(U) satisfies βU(g) = 0, then there exists an open covering Ujj∈Jof U and elements fj ∈ F(Uj), such that αUj

(fj) = g|Uj.

The homomorphisms α and β induced group homomorphisms at the stalk level,

Fxαx→ Gx βx→ Hx. The above definition is equivalent to requiring that, for each x in

X, this sequence is exact; that is, ker(βx) = im(αx)

1.5.20. Lemma. If 0 → F α→ G β→ H is exact (where the first homomorphism,

0 → F is the trivial homomorphism) then for every open subset U of X, the sequence

0 → F(U)αu→ G(U)

βU→ H(U) is exact.

Proof. Step 1: α : F(U) → G(U) is injective. This fact follows immediately

from the definition of exact sequence. Assume that f ∈ F(U) is mapped to the 0

element by αU . From (2) in 1.5.19 we have that there exists a covering Ujj of U ,

such that f |Uj= 0 (since the first sheaf is trivial). But this implies that f = 0.

Step 2: α(F(U)) ⊂ ker(βU). This is simply condition (1) in the definition of exact

sequence.

Step 3: ker(βU) ⊂ α(F(U)). Let β(g) = 0 for some g ∈ G(U). We have a

covering Ujj of U , and elements fj ∈ F(Uj), such that α(fj) = g|Uj. But then

α(fj − fk) = g − g = 0 on Uj ∩ Uk; since α is injective by the first step in this


proof, we have that fj = fk. Thus there exists f ∈ F(U) such that f |Uj= fj, which

implies α(f) = g.

The first step in the above proof will be frequently used, so we write it as a separate

result for easier reference.

Corollary. If 0 → F α→ G is exact, then for any open set U ⊂ X, the mapping

αU : F(U)α→ G(U) is injective.

1.5.21. A sheaf homomorphism α : F → G induces mappings between

cohomology groups in a natural way. For the 0-th cohomology groups, since

H0(X,F) = F(X) and H0(X,G) = G(X), the mapping αX : F(X) → G(X) can be

considered as a mapping in cohomology, α0 = αX : H0(X,F) → H0(X,G).

To make a similar contruction with 1st cohomology groups, we consider a class

ξ in H1(X,F). Choose a cochain to represent this class, ξ = [(fjk)], in some

open covering U = Ujj∈J of X, with (fjk) ∈ Z1(U ,F). It is easy to check that

(αUj∩Uk(fjk)) is an element of Z1(U ,G). Moreover, if (fjk) = (fj−fk) is a coboundary

we have (α(fjk)) = (α(fj)−α(fk)) is also a coboundary. Thus we obtain a mapping

α1U : H1(U ,F) → H1(U ,G) defined by the expression α1 ([(fjk)]) = [(α(fjk))]. Is it

easy to chek that if V is a covering of X finer that U , then φUV α1

U = α1V φU

V . Thus

we can define mapping α1 : H1(X,F) → H1(X,G) by α1(ξ) = [(α(fjk))].

Assume now that 0 → F α→ G β→ H → 0 is an exact sequence. From the above

constructions we get two sequences

0 → H0(X,F)α0

→ H0(X,G)β0

→ H0(X,H),

and

H1(X,F)α1

→ H1(X,G)β1

→ H1(X,H).

We want to have a homomorphism δ∗ : H0(X,H) → H1(X,F) connecting them.

To do that, consider an element h of H0(X,H) = H(X). Since the sheaf sequence

0 → F αX→ G βX→ H → 0 is exact (lemma 1.5.20) and h is mapped to the 0 element

(the last homomorphism is trivial), we have a covering U = Ujj∈J , and elements

gj ∈ G(Uj), such that β(gj) = h|Uj. Clearly β(gj − gk) = 0, so there exist fjk ∈


F(Uj ∩ Uk) with α(fjk) = gj − gk. It is not difficult to see that (fjk) is a cocycle:

α(fjk + fkl + flj) = α(fjk) + α(fkl) + α(flj) = gj − gk + gk − gl + gl − gj = 0,

and therefore fjk + fkl + flj = 0, since α is injective (corollary 1.5.20). One can

check similarly that the cohomology class of (fjk) does not depend on the choices

made in its construction. We then set δ∗(h) = f , where f = [(fjk)] ∈ H1(X,F).

1.5.22. Theorem. Assume 0 → F α→ G β→ H → 0 is an exact sequence.

Then the sequence

0 → H0(X,F)α0

→ H0(X,G)β0

→ H0(X,H)δ∗→ H1(X,F)

α1

→ H1(X,G)β1

→ H1(X,H)

is exact.

Proof. Step 1. The first terms of the sequence are

0 → F(X)α0

→ G(X)β0

→ H(X),

which is exact by lemma 1.5.20.

Step 2: im(β0) ⊂ ker(δ∗). Let h ∈ H(X) and g ∈ G(X) such that β0(g) = h.

We have to show that δ∗(h) = 0, in H1(X,F). In the construction of δ∗ we take

elements gj ∈ G(Uj) (for some covering of X), such that β(gj) = h|Uj. Clearly in

this situation we can take gj = g|Uj, and then we get α(fjk) = g − g = 0. Since α is

injective we have fjk = 0, therefore δ∗(h) = f = 0 (with the above notation).

Step 3: ker(δ∗) ⊂ im(β0). Suppose δ∗(h) = ([(fjk)]) splits; that is, [(fjk)] =

[(fj−fk)], for some 0-cochain (fj). Let (gj) be the 0-cochain used in the construction

of δ∗(h); then gj −α(fj) = gk−α(fk) on (Uj ∩Uk). By the definition of sheaf, there

exists g ∈ G(X) satisfying g|Uj= gj − α(fj). Then β(g)|Uj

= β(gj) − β(α(fj)) =

β(gj) = h|Uj, or, in other words, β(g) = h.

Step 4: im(δ∗) ⊂ ker(α1). If h ∈ H(X) satisfies δ∗(h) = [(fjk)], then α(fjk) =

gj − gk, that is, (α(fjk)) splits.

Step 5: ker(α1) ⊂ im(δ∗). Assume α(fjk) = gj − gk. Then β(gj) = β(gk) +

β(α(fjk)) = β(gk), so there exists h ∈ H(X) with h|Uj= β(gj). Clearly δ∗(h) =

[(fjk)].


Step 6: im(α1) ⊂ ker(β1). This follows from the exactness of the sequence

F(Uj ∩ Uk) α→ G(Uj ∩ Uk) β→ H(Uj ∩ Uk).

Step 7: ker(β1) ⊂ im(α1). Assume that (gjk) is a cocycle in X (with coefficients in

the sheaf G), such that β(gjk) = hj −hk for some 0-cochain (with coefficients in H).

Since G → H → 0 is exact, by shrinking the covering if necessary, we can further

assume that hj = β(gj), for elements hj ∈ H(Uj). Then

β(gjk − (gj − gk)) = 0,

so there exists a 1-cochain (fjk) satisfying α(fjk) = gjk − gj + gk. Clearly α maps

(fjk + fkl + flj) to 0, and therefore, by the injectivity of αUj∩Uk∩Ul, the cochain (fjk)

is a cocycle. Moreover, we have

α1([(fjk)]) = [(α(fjk))] = [(gjk − gj + gk)] = [(gjk)],

which completes the proof of the theorem.

1.5.23. Corollary. In the hypothesis of the above theorem, if H1(X,G) = 0,

then H1(X,F) ∼= H(X)/β (G(X)).

Proof. From the previous theorem we have that the following sequence is exact,

G(X)β→ H(X)

α1

→ H1(X,F) → 0.

So α1 is surjective, and the result follows.

The de Rham theorem

1.5.24. We have seen that the exterior derivative can be considered as a sheaf

homomorphism from S (smooth functions) to S1 (1-forms). If a function f satisfies

df = 0, then it is locally constant. Thus C, the sheaf of locally constant functions

with values in the complex numbers, is the kernel of this sheaf homomorphism. Let

S1c denote the sheaf of closed forms. We claim that the sequence

0 → C → S d→ S1c → 0


is exact, where the mapping C → S is the natural inclusion. The only step that

requires some work is to show that d is locally surjective. In other words, we have

to prove that any closed form is locally exact. This is precisely the statement of

Poincare’s lemma (1.4.8), which we prove next.

Consider a closed form ω = fdx+ gdy on the unit disc D. Since dω = 0 we have

fx(x, y) = gy(x, y). Define a function F : D → C by the expression

F (x, y) =

∫ 1

0

(x f(tx, ty) + y g(tx, ty)) dt, (x, y) ∈ D.

For the reader used to path integration the function F can be written as

F (x, y) =

∫

c

(f(z) + g(z)) dz,

where c is the path with image the straight line from 0 to (x, y); that is, c(t) =

(tx, ty) : [0, 1] → D.

Clearly F is a smooth function; we will show that dF = ω. Observe first that

∂f(tx, ty)

∂t= x

∂f(tx, ty)

∂x+ y

∂f(tx, ty)

∂y;

and therefore, using the equality fx = gy we get

t

(∂f(tx, ty)

∂t

)= tx fx(tx, ty) + ty gx(tx, ty).

If we now compute the derivative of F we see

∂F (x, y)

∂x=

∫ 1

0

∂

∂x(x f(tx, ty) + y g(tx, ty)) dt =

=

∫ 1

0

(f(tx, ty) + tx fx(tx, ty) + ty gx(tx, ty)) dt =

=

∫ 1

0

(f(tx, ty) + tft(tx, ty)) dt =

∫ 1

0

∂

∂t(tf(tx, ty)) dt

=tf(tx, ty)|t=1t=0 = f(x, y).

A similar argument shows that ∂F/∂y = g, so dF = ω, as claimed.

The group S1c (X)/dS(X) is known as the de Rham group of X, and denoted

by H1dR(X).

Theorem. The groups H1dR(X) and H1(X,C) are isomorphic.


Proof. It follows from the above computations, corollary 1.5.23 and the fact

that H1(X,S) = 0 (1.5.12).

Remark. It is possible to prove that in each class of H1dR(X) there is a unique

harmonic form. This is the content of the Hodge theorem, which we will not show

here. See [7] for a classical approach, or [10] for a modern development.

The 2nd Cohomology Group

1.5.25. In §2.5 we will use 2nd cohomology groups as well. Since the con-

structions and proofs are similar to what we have done so far in this section we will

describe them only briefly. The interested reader can fill in the details.

The coboundary homomorphism δ : C2(U ,F) → C3(U ,F) is given by δ(fijk) =

(gijkl), where

gijkl = fjkl − fikl + fijl − fijk

on Ui ∩Uj ∩Uk ∩Ul. The groups of cocycles and coboundaries are defined in a way

similar to the 1-cochains case. Lemma 1.5.7 and its proof generalise to this setting,

where the operator K3 : C3(U ,F) → C2(V,F) is given by

K3(fijkl) = fφ(α)ψ(α)ψ(β)ψ(γ) − fφ(α)φ(β)ψ(β)ψ(γ) + fφ(α)φ(β)φ(γ)ψ(γ) .

We leave it to the reader to finish the construction of H2(X,F).

1.5.26. The long exact sequence of 1.5.22 can be extended to include the second

cohomology groups.

Theorem. Assume 0 → F α→ G β→ H → 0 is an exact sequence. Then there

exists a group homomorphism γ∗ : H1(X,H) → H1(X,F) such that the following

sequence is exact:

0 →H0(X,F)α0

→ H0(X,G)β0

→ H0(X,H)δ∗→ δ∗→ H1(X,F)

α1

→ H1(X,G)β1

→β1

→H1(X,H)γ∗→ H2(X,F)

α2

→ H2(X,G)β2

→ H2(X,H).

CHAPTER 2

Compact Riemann surfaces

2.1 Divisors 58

2.2 Dolbeault’s Lemma and Finiteness Results 62

2.3 The Riemann-Roch Theorem 68

2.4 Line bundles and Divisors 71

2.5 Serre Duality 81

2.6 Applications of the Riemann-Roch Theorem 95

2.7 Projective embeddings 99

2.8 Weierstrass Points and Hyperelliptic Surfaces 106

2.9 Jacobian Varieties of Riemann Surfaces 115

57

58 2. COMPACT RIEMANN SURFACES

In this chapter we study compact Riemann surfaces in some detail and obtain

several important results. We start (§2.1) with the definition of divisor: loosely

speaking a divisor is a finite set of points of a surface to which certain integral

weights (numbers) are assigned. We are interested on finding functions and forms

whose zeroes and poles are determined by a divisor. This question is naturally

related (and in a some sense a generalisation) to the problem of finding non-constant

meromorphic functions on surfaces. The dimensions of the spaces of functions and

forms associated to a divisor are given by the Riemann-Roch theorem, one of the

most important results in the theory of compact Riemann surfaces. We prove a first

version of this theorem in §2.3. A second version is proved in §2.6; for that we need

to introduce the concepts of line and vector bundles on Riemann surfaces (§2.4) and

prove another important theorem, the Serre Duality theorem, which relates certain

spaces of functions and forms. We show the power of these result by giving a few

applications in the rest of the sections of the chapter. In particular we proved that

there exists only one Riemann surface structure on the Riemann sphere (2.6.8); a

generalisation of this result says that any surface can be considered as a subset of

a projective space (§2.7). Using the Riemann-Roch and Serre Duality theorems we

construct in §2.9 certain higher dimensional manifolds, called Jacobian varieties,

which are a generalisation of tori (1.3.6), and a mapping from a Riemann surface to

its Jacobian variety, the Abel-Jacobi mapping. We will use this mapping to show

that any surface of genus 1 is indeed is a torus.

2.1. Divisors

This short section introduces the concept of divisor on a compact Riemann sur-

face X: a divisor D is simply a finite set of points of X with certain integral

“weights”. We associate two sheaves of vector spaces to any divisor, consisting of

meromorphic functions and forms, whose poles and zeroes have orders bounded by

D. The Riemann-Roch theorem, proved in a later section, relates the dimensions of

these two spaces.

2.1. DIVISORS 59

2.1.1. Definition. A divisor D on a compact Riemann surface X is a formal

finite sum of the type

D =m∑

j=1

nj pj,

where nj ∈ Z, m is a non-negative integer and pj are points of X.

The set of divisors on X, Div(X), is an abelian group where the operation is the

“natural” formal addition:

(n1p1 + · · ·+ nmpm) + (r1q1 + · · ·+ rsqs) = n1p1 + · · ·+ nmpm + r1q1 + · · ·+ rsqs.

In another terminology, a divisor is an element of the free abelian group generated

by the points of X.

For a point p in X we will denote by D(p) the “value” of D at p, namely D(p) = nj

if p = pj , for some j = 1, . . . , m and D(p) = 0 otherwise.

The zero divisor (m = 0) D = 0 is defined by D(p) = 0 for all p ∈ X. It is the

identity element of Div(X).

2.1.2. Definition. A divisor D is called effective, denoted by D ≥ 0, if

nj ≥ 0, for all j = 1 . . . , m. We write D1 ≥ D2, for divisors D1 and D2, if D1 −D2

is effective.

2.1.3. If f : X → C is a non-identically zero meromorphic function we can

assign to it a divisor div(f) by considering the order of f at the points of X. More

precisely, given a local coordinate z, defined on a neighbourhood of p, the function

f has a Laurent power series expansion of the form f(z) =∑∞

j=n cj (z − z(p))j ,

with cn 6= 0. We set ordp(f) = n, and define div(f) =∑

p∈X ordp(f) p. Observe

that ordp(f) 6= 0 only when p is a zero or a pole of X, so this sum is finite (X is

compact). If p is a zero of f , then ordp(f) is the order of that zero. For the case of

a pole one has to be slightly careful; ordp(f) is a negative number, the opposite of

what is usually known as the order of a pole (1.1.8). To be more precise, consider

for example a polynomial p(z) = a(z − z1) · · · (z − zn) defined on C. Its divisor is

given by D = div(p) = z1 + · · ·+ zn−n∞ (the sum here takes place in the group of

divisors, not in the complex plane!) So D(∞) = −n, although one usually speaks of

∞ as a pole of order n. We hope this small matter of terminology does not confuse


the reader. We have that the divisor of a meromorphic function is the (formal) sum

of the zeroes and poles of the function, counted with multiplicities (and signs!).

We define in a similar way the divisor of a meromorphic 1-form ω. If this form is

given locally by ω = fdz, we set ordp(ω) = ordp(f), for p a point in the domain of

definition of the coordinate z. If t is another coordinate (defined in a neighbourhood

of p) and ω = gdt, then f = g dtdz

. Recall that dtdz

means ∂(tz−1)∂z

(z(q)), and therefore

dtdz

6= 0, since z and t are local coordinates. This shows that ordp(f) = ordp(g); or,

in other words, the order of ω at a point p, ordp(ω), is well defined. We can then

define the divisor of ω by div(ω) =∑

p∈X ordp(ω) p.

2.1.4. Definition. Two divisors D1 and D2 are linearly equivalent, D1 ∼D2, if there exists a meromorphic function f such that (f) = D1 −D2.

Consider on the Riemann sphere two divisors D1 = z1 and D2 = z2, where z1 and z2

are points of C. If z1 6= ∞ and z2 6= ∞ then the divisor of the function f(z) = z−z1z−z2

is equal to div(f) = z1 − z2, so D1 ∼ D2. If z1 = ∞ we take f(z) = 1z−z2 , while we

consider the function f(z) = z − z1 if z2 = ∞. In any of these two last cases we

also have that div(f) = D1 −D2. Thus we have proved that D1 and D2 are linearly

equivalent. See the comments after the next definition for a more general statement.

If ω1 and ω2 are two meromorphic forms it is easy to see that the quotient ω1/ω2

is a meromorphic function on X: suppose that ωj = fjdz on some local coordinate

z, where fj are meromorphic functions (j = 1, 2). Then we have ω1

ω2= f1

f2. If we have

another local coordinate, say t (whose domain of definition intersect that of z), then

ωj = gjdt, and the relation fj = gjdtdz

holds. We have

f1

f2

=g1(dt/dz)

g2(dt/dz)=g1

g2

,

so ω1/ω2 does not depend of the choice of local coordinates. This shows that the

divisors of any two meromorphic forms are linearly equivalent.

2.1.5. Definition. The degree of a divisor D =∑m

j=1 nj pj is defined by the

integer

deg (D) =m∑

j=1

nj .

2.1. DIVISORS 61

It is easy to prove that two divisors on the Riemann sphere are linearly equivalent

if and only if they have the same degree (a generalisation of the above argument);

see exercise 42.

2.1.6. A meromorphic function has as many zeroes as poles, counted with mul-

tiplicities (1.3.11) and therefore deg (div(f)) = 0. It follows that linearly equivalent

divisors have the same degree. In particular, the divisors of two meromorphic forms

have the same degree (see 2.5.19).

Definition. A divisor is called principal if it is the divisor of a meromorphic

function, and canonical if it is the divisor of a meromorphic 1-form.

The divisor class group is the quotient of the group of divisors by the subgroup

of principal divisors, Div(X)/DivP (X). The canonical class is the class of any

canonical divisor in the divisor class group. Since the degree of a canonical divisor

does not depend of the form we have that the degree of the canonical class is well

defined; in 2.5.19 we will show that this degree is equal to 2g−2, where g is the genus

of X. We will denote by KX (or K) both the canonical class and a representative

of it.

2.1.7. We associate a sheaf of meromorphic functions to a divisor D by setting

O(D)(U) = f : U → C; f meromorphic and div(f) ≥ −D on U or f ≡ 0,

where U is an open set ofX. For the zero divisor, D = 0, we will simplify notation by

writing O (instead of O(0)). We have H0(X,O) is the set of holomorphic functions

on X; since X is compact, any holomoprhic function is constant, and thus we have

that H0(X,O) is isomorphic to C.

If D(x) ≥ 0, and div(f) ≥ −D, then f can have a pole of order (as pole!) at most

D(x) at the point x. On the other hand, if D(x) < 0, f must have a zero of order

at least D(x) at x. For example, if z0 ∈ C, and D = z0, the set O(D)(C) consists of

the holomorphic functions on C (i.e. the constant functions), and the meromorphic

functions with a simple pole at z0. These functions form a vector space of dimension

2; a basis is given, for example, by f1 ≡ 1, f2(z) = 1/(z − z0), when z0 6= ∞, and


f1 ≡ 1, f2(z) = z if D = ∞.

Proposition. If deg (D) < 0 then H0(X,O(D)) = 0.

Proof. Since div(f) has 0 degree and deg (−D) is positive we cannot have

div(f) ≥ −D.

We associate a sheaf of meromorphic forms on X to the divisor D by the expres-

sion

Ω(D)(U) = ω ∈ M1(U); div(ω) ≥ −D,

for U ⊂ X open. We will simplify notation and write Ω(U) when we consider the

zero divisor. The space H0(X,Ω) is the set of holomorphic 1-forms of X.

2.1.8. One can generalise definition 2.1.1 to non-compact surfaces in the fol-

lowing way. A divisor on a Riemann surface X is a map D : X → Z such that for

any compact subset K of X, the set

x ∈ X; D(x) 6= 0

if finite. Clearly both definitions agree if X is compact.

2.2. Dolbeault’s Lemma and Finiteness Results

The solutions of the Cauchy-Riemann equations, fz = 0, are the holomorphic

functions. In this section we study the inhomogeneous equation, fz = g, where g

is a smooth function, and show that it always has a solution. In the second part

of the section we use this fact to prove that the cohomology groups Hn(X,O) are

finite dimensional vector spaces (for X compact and n = 0, 1).

Dolbeault’s Lemma

2.2.1. We start with the case of g having compact support.

Theorem. Let g : C → C be a smooth function with compact support. Then

there exists a smooth function f : C → C, such that fz = g.

Proof. Define f by the integral

f(z) =1

2πi

∫

C

g(w)

w − zdw dw.

2.2. DOLBEAULT’S LEMMA AND FINITENESS RESULTS 63

We first need to prove that f is well defined; that is, the integral is finite. Write

w = z + reiθ. Then

f(z) =1

2πi

∫

C

g(z + reiθ)

reiθr dr dθ =

−1

π

∫

C

g(z + reiθ) e−iθ dr dθ.

In the above computation we have used the identity dw ∧ dw = −2i r dr ∧ dθ. The

function g has compact support so it is bounded and therefore the last integral in

the above expression is finite. This shows that f is well defined; moreover we can

interchange derivatives and integration when computing ∂f . Using this one easily

sees that

∂f

∂z(z) =

−1

π

∫

C

∂g(z + reiθ)

∂ze−iθ dr dθ.

Write ξ = reiθ; then

∂f

∂z(z) =

1

2πlimr→0

∫

|ξ|≥r

∂g(z + ξ)

∂z

r

ξdr dθ.

Since the variables ξ and z are interchangeable in the function g(z + ξ) we have

∂g

∂z(z + ξ)

1

ξ=∂g

∂ξ(z + ξ)

1

ξ=

∂

∂ξ

(g(z + ξ)

ξ

).

Substituting this identity in the previous integral we obtain

∂f

∂z(z) =

1

2πilimr→0

∫

|ξ|≥0

∂

∂ξ

(g(z + ξ)

ξ

)dξ dξ =

−1

2πilimr→0

∫

|ξ|≥rd

(g(z + ξ)

ξdξ

)

= limr→0

∫

|ξ|=r

g(z + ξ)

2πiξdξ.

In the last equality we have used Stokes’ theorem (1.4.22). The change of sign in the

integral is due to the orientation of the boundary of the integration domain when

applying Stokes’ theorem. If we put ξ = reiθ in the last integral, we get

∂f

∂z(z) =

1

2πlimr→0

∫ θ=2π

θ=0

g(z + reiθ) dθ.

The last integral is nothing but the average of the function g on the circle of centre

z and radius r. Since g is uniformly continuous these averages converge to the value

of g at the point z (see exercise 43 for hints). In other words

∂f

∂z(z) = g(z).


2.2.2. By covering C with an increasing sequence of discs we can remove form

the previous result the condition of compactness of the the support of g.

Theorem (Dolbeault). Let g : C → C be a smooth function. Then there exists

a smooth function f : C → C with fz = g.

Proof. As remarked before the statement of the theorem the idea of the proof

is to cover C by an increasing sequence of compact sets, on which we can solve the

inhomogeneous Cauchy-Riemann equation; the limit of these “compact” solutions

will be the desired function.

We start by setting Bn = z ∈ C; |z| < n. Since C is a normal space ([16,

Theorem 2.3 in pg. 198 and Urysohn Lemma in pg. 207]) there exist smooth

functions φn, with compact support in Bn+1, such that φn ≡ 1 on the disc z ∈C; |z| < n+ 1

2. (Observe that this disc contains the compact set Bn). Let gn = φn g.

These functions have compact support, and therefore we can find smooth functions

hn, with support contained in Bn+1, such that ∂hn = gn (in Bn+1 or C). The

functions hn+1 − hn are holomorphic on a neighbourhood of Bn, so they have a

power series expansion that converges uniformly on Bn. Truncating that power

series we get a polynomial pn, such that |(hn+1 − hn− pn)(z)| < 2−n, for all z in Bn

(this is a trivial case of Runge’s Theorem, [20, theprem 113.7, pg. 290]). Define a

function f by the expression

f = hn +

( ∞∑

m=n

(hm+1 − hm − pm)

)− (p1 + · · · + pn−1) .

It is easy to see (exercise 44) that the function f is well defined. The right hand

side of the above equality is a uniformly convergent series of holomorphic functions,

so f is holomorphic and satisfies ∂f = ∂hn = g on Xn.

Observe that solutions of the equation ∂f = g are not unique: if f is one solution

so will be f + h, where h is a holomorphic function (since ∂h = 0).

2.2.3. Corollary (of the proof). The above theorem holds for the case

of a disc; that is, when g is defined on z ∈ C; |z| < R (R > 0).

Actually the result holds for any open set of the complex plane, see [19].


2.2.4. Suppose U is a domain of C biholomorphic to a disc; that is, there exists

a disc DR = z ∈ C; |z| < R and a biholomorphic mapping f : U → DR. Then

the inhomogeneous Cauchy-Riemann equation can be solved on U for any smooth

function g : U → C. The proof, which is a simple application of the formulæ 1.1.3,

is left as an exercise to the reader.

2.2.5. Corollary. The equation ∆f = g has solution in C for any smooth

function g.

Proof. Let h and k be smooth functions such that hz = g and kz = h. Then

f = i2k satisfies ∆f = g (use the identity ∆ = 2 i ∂ ∂)

2.2.6. Using Dolbeault’s lemma we can compute the dimension of certain co-

homology groups.

Theorem. If X = C or X = z ∈ C; |z| < R then H1(X,O) = 0.

Proof. Let (fjk) ∈ Z1(U ,O) for some covering U = Ujj∈J . Since holomorphic

functions are smooth and H1(X,S) = 0 (1.5.12) we have functions gj ∈ S(Uj), such

that fjk = gj−gk. This implies that ∂gj = ∂gk, so there exists a (smooth) function h

on X satisfying ∂gj = h|Uj. Let g be a solution of ∂g = h, and set fj = gj−g. Then

∂fj = 0, i.e. fj is holomorphic, and fjk = fj − fk. This shows that any arbitrary

cocycle splits and therefore H1(X,O) = 0.

Because of 2.2.4 the above theorem also holds if X is biholomorphic to a disc.

2.2.7. Corollary. For a Riemann surface X the following sequence is exact:

0 → O → S ∂→ S(0,1) → 0.

Proof. Since the kernel of the operator ∂ is the set of holomorphic functions,

exactness at the first step is satisfied. The question at the next step is local; we

need to show that any form of type (0, 1) is locally exact. Thus we can assume that

X is a disc in the complex plane. If ω is of type (0, 1) (in a disc), then ω = g dz,


for some smooth function g. Clearly the form ω is exact if and only if we can find

a function f such that ∂f = g. But this is simply Dolbeault’s lemma (2.2.3).

Finiteness Results

2.2.8. The sheaf O on a Riemann surface X has the structure of a vector space

(i.e. O(U) are vector spaces) and therefore the cohomology groups Hn(X,O) are

vector spaces. In the next paragraphs we will show that these spaces are finite

dimensional, for n = 0, 1 and X a compact surface. The case of n = 0 is just a

simple consequence of the Maximum Modulus Principle (1.1.9 and 1.3.8).

Theorem. For a compact Riemann surface X, H0(X,O) = C.

2.2.9. The proof for H1(X,O) for the case when X is the Riemann sphere is

an easy consequence of basic results of Complex Analysis.

Theorem. H1(C,O) = 0.

Proof. Consider the standard covering of C given by U1 = C and U2 = C\0(1.3.5). Both sets U1 and U2 are biholomorphically equivalent to the complex plane,

so H1(Uj ,O) = 0, j = 1, 2; or, in other words, the covering U1, U2 is a Leray

covering for the sheaf O. If we take a cocycle, (fjk)j,k=1,2 (with coefficients in the

sheaf O), since f11 = f22 = 0 and f12 = −f21, the only function we need to consider

is f12. Let f12(z) =∑+∞

−∞ aj zj be a Laurent power series representation of this

function. The functions f1(z) =∑∞

j=0 aj zj and f2(z) = −∑−∞

j=−1 aj zj are elements

of O(U1) and O(U2), respectively, satisfying f12 = f1 − f2.

2.2.10. Theorem. If X is a compact Riemann surface then H1(X,O) is finite

dimensional.

Proof. Let Dr denote the disc in the complex plane centred at 0 and radius

r, where r > 0. For every point p ∈ X there exists a coordinate patch (U, z), such

that zp(p) = 0 and zp(U) = D2. The collection z−1p (D1/2), as p varies over all

the points of X and z is as above, forms an open covering of X and therefore there

exists a finite subcover W. To simplify notation we will write z1, . . . , zn for the

local coordinates of this subcover. Thus W = Wj = z−1j (D1/2)nj=1 is our covering.


The collections V = Vj = z−1j (D1)nj=1 and U = Uj = z−1

j (D2)nj=1 are also open

coverings of X. Moreover, we have W < V < U . Observe that Wj (respectively Vj)

has compact closure contained in Vj (respectively Uj). Since the open sets of these

three coverings are biholomorphically equivalent to discs (in C) we have that W, Vand U are Leray coverings of X for the sheaf O.

Let (fjk) be an element of C0(U ,O), and denote by gjk the restriction of fjk to

Vjk. The functions gjk are bounded, since Vjk has compact closure contained in Ujk.

Assume that (gjk) splits, that is, gjk = gj − gk. We claim that gj is bounded in Vj.

To prove it consider a point p of the boundary of Vj; then there exists an open set Vl

such that x ∈ Vl. Let A be a neighbourhood of x with A compact, contained in Vl.

Then the functions gl and gjl are bounded in A∩ Vj ∩ Vl, which is a neighbourhood

of x. Since cj = gjl + cl we have that cj is bounded in that neighbourhood. In other

words, every point of ∂Vj has a neighbourhood in Vj where cj is bounded. By the

Maximum Modulus Principle (1.1.9) the function cj is bounded in Vj.

The above argument shows that to compute H1(X,O) we can use bounded

cochains. The advantage of this is that the space of bounded cochains is a Banach

space. More precisely, if we consider η = (fj) ∈ C0(V,O), where the functions fj ’s

are bounded, we can define a norm by ||η|| =∑n

j=1 ||fj||∞ (||fj||∞ = sup|fj(z)|; z ∈Vj). A similar definition gives us a norm in the space of 1-cochains. In what follows

we will assume all cochains to be bounded.

We now define two mappings

δ : C0(W,O) → C1(W,O) and β : Z1(V,O) → Z1(W,O)

as follows. The mapping δ is the coboundary homomorphism while β is the mapping

induced by restriction (recall that W < V). Both mappings are clearly continuous

with respect to the topology induced by the norms introduced above. Moreover by

Montel’s theorem (1.1.13) we have that β is a compact mapping.

Set:

ψ : C0(W,O) × Z1(V,O) →Z1(WO)

(η, ξ) 7→δ(η) + β(ξ).


Since H1(X,O) = H1(W,O) = H1(V,O) the mapping β is surjective. It follows

that ψ is surjective as well.

The mapping

φ : C0(W,O) × Z1(V,O) →Z1(W,O)

(η, ξ) 7→ψ((η, ξ)) − β((η, ξ)) = δ(η).

is the difference of a surjective mapping and a compact one. Its image is the space

of coboundaries, B1(,W,O). The quotient Z1(W,O)/B1(W,O) is nothing but

H1(X,O); by Schwartz’s theorem (1.1.25) it has finite dimension.

2.2.11. Definition. If X is a compact Riemann surface, the dimension of

H1(X,O) is called the arithmetic genus of X.

2.3. The Riemann-Roch Theorem

Given a divisor D on a surface we have associated to it the shear O(D) consisting

of meromorphic functions whose zeroes and poles have orders “bounded” by D. If

this sheaf is not empty for certain divisors then we can easily prove the existence of

non-constant meromorphic functions on surfaces. It is thus of interest to know the

dimension of the cohomology groups of O(D). The Riemann-Roch theorem proves

that the difference dim H0(X,O(D))−dim H1(X,O(D)) is a constant that depends

only on the degree of D and the arithmetic genus of X. We prove this result here

and use it to show the existence of meromorphic functions on compact surfaces by a

proper choice of the divisor D. We also prove that any surface of arithmetic genus

0 must be biholomorphic to the Riemann sphere C. In § 2.6 we will show that the

arithmetic genus of a compact surface is equal to its topological genus; these two

facts show that the Riemann sphere admits only one Riemann surface structure (up

to biholomorphisms).

2.3.1. Let X be a compact surface x a point of it. The skyscraper sheaf Cp

is defined, on an open set U ⊂ X, by

Cx(U) =

C, if p /∈ U,

0, otherwise,

2.3. THE RIEMANN-ROCH THEOREM 69

with the obvious restriction mappings. Clearly H0(X,Cp) = Cp(X) = C. To

compute the first cohomology group consider an open covering of X, say U = Ujj.Choose an index j0 such that p ∈ Uj0 , and define a new covering of X, V = Vjjby setting

Vj =

Uj0 , if j = j0,

Uj\p, otherwise.

Since p does not belong to Vj ∩ Vk, for j 6= k, we have H1(V,Cp) = 0. The natural

mapping H1(U ,Cp) → H1(V,Cp) is injective; therefore H1(U ,Cp) = 0. Since U is

an arbitrary covering of X we obtain H1(X,Cp) = 0.

2.3.2. Let D be a divisor on X. Fix a point p ∈ X; we construct a sequence

(4) 0 → O(D) → O(D + p)β→ Cp → 0

as follows. If f ∈ O(D), then (f) ≥ −D ≥ −D − p, so O(D) is a subsheaf of

O(D + p). To define β, consider first the case of a an open set U such that p /∈ U ;

then Cp(U) = 0 so β ≡ 0. On the other hand, if p ∈ U , let f be a meromorphic

function with f ∈ O(D+p)(U); choose a local coordinate z around x with z(p) = 0.

The function f will have a power series expansion of the form f(z) =∑∞

j=−n−1 aj zj

near p. We set β(f) = a−n−1, where D(p) = n (see the remark below). If Cp(U) is

not trivial then the kernel of β consists precisely of the functions that have a pole

of order at most n at p; that is, O(D)(U). On the other hand, if Cp(U) = 0, then

x /∈ U , so D|U = (D + p)|U (both divisors have the same values at all points of U)

and O(D)(U) = O(D + p)(U). Thus (4) is exact.

Remark. It can be easily checked that if w is another local coordinate on X

with w(p) = 0, and we write f(w) =∑∞

j=−n−1 bjwj , then b−n−1 = a−n−1. Thus the

mapping β above is well defined.

2.3.3. Theorem (Riemann-Roch). Let X be a compact Riemann surface and

D a divisor on X. Then the spaces H0(X,O(D)) and H1(X,O(D)) are finite di-

mensional and

dim H0(X,O(D)) − dim H1(X,O(D)) = 1 − g′ + deg (D),


where g′ is the arithmetic genus of X, i.e. g′ = dim H1(X,O).

Proof. Fix a point p ∈ X as above. The long exact sequence corresponding to

(4) is

0 →H0(X,O(D)) → H0(X,O(D + p)) → C → H1(X,O(D)) →

→H1(X,O(D + p)) → 0.(5)

We can split (5) in two sequences given by

0 →H0(X,O(D)) → H0(X,O(D + p)) → V → 0

0 →W → H1(X,O(D)) → H1(X,O(D + p)) → 0,

where V is the image of the mapping H0(X,O(D + p)) → C, and W = C/V . Both

the sequences are clearly exact because of the choice of V and W . Assume now that

the result is true for one of the two divisors, D or D + p. Since V and W have

finite dimension we have that two spaces in each of the above sequence are finite

dimensional, and thus all spaces must have finite dimension. From the exactness of

those sequences we see that

dim H0(X,O(D)) + dim V = dim H0(X,O(D + p))

dim H1(X,O(D + p)) + dim W = dim H1(X,O(D)).

These two equations together with the equality dim V + dim W = dim C = 1 give

dim H0(X,O(D)) − dim H1(X,O(D)) =

dim H0(X,O(D + p)) − dim H1(X,O(D + p)) − 1

But deg (D) = deg (D + p) − 1, so the above identity is equivalent to the following

dim H0(X,O(D)) − dim H1(X,O(D)) − deg (D) =

dim H0(X,O(D + p)) − dim H1(X,O(D + p)) − deg (D + p).

Since we are assuming that the theorem is true for one of the two divisors, one side

of the above equation is equal to 1 − g′, and so is the other side.

To complete the proof we need three observations:

(1) ifD is the zero divisor then dim H0(X,O) = dim C = 1 and dim H1(X,O) =

g′, so the theorem holds in this case.

(2) From the above arguments we have that if the theorem is true for a divisor

2.4. LINE BUNDLES AND DIVISORS 71

E then it is also true for the divisors E+ p and E− p, where p is an arbitrary point

of X.

(3) Starting with the zero divisor we can obtain any divisor in a finite number

of steps by adding and subtracting points.

2.3.4. Corollary. If X is a compact Riemann surface, then there exists a

non-constant meromorphic function f : X → C.

Proof. Consider the divisor D = (g+1) p, for some point p of X. The constant

functions form a one-dimensional subspace of H0(X,O(D)). On the other hand,

from the Riemann-Roch theorem we obtain

dim H0(X,O(D)) ≥ 2,

so there must exist a non-constant function in O(D).

2.3.5. Corollary. If X is a compact surface of arithmetic genus 0 then X

is biholomorphic to C.

Proof. From the previous result we have that if D = p then there exists a

non-constant meromorphic function f : X → C, with f ∈ H0(X,O(D)). Because

of the choice of D the function f can have at most a simple pole at p. On the other

hand, since f is not constant, it must have a pole at p (the only allowed singularity).

Thus f has degree 1 (see 1.3.11); that is, it is biholomorphic.

2.4. Line bundles and Divisors

The purpose of this section is to introduce the concept of vector bundles on

Riemann surfaces and prove some results that we need in later sections. Informally

speaking, a vector bundle is just a way of assigning a vector space to each point of

a surface such that, as we move “holomorphically” on the surface the corresponding

vector spaces change also “holomorphically”. The space of line bundles (vector

bundles of dimension 1) is called the Picard group of the surface; we show that this

group is isomorphic to H1(X,O∗). Using vector bundles we give a new proof of

the existence of non-constant meromorphic functions on compact surfaces. We will

also show that there is a natural way of associating a line bundle to divisors on a


surface X. We will study the relationship between the sheaf O(D) and isomorphic

line bundles (we need those results for the next section).

Vector Bundles on Riemann surfaces

2.4.1. We begin this section by extending the notions of Riemann surfaces and

holomorphic mappings to the setting of several complex variables. Assume Ω is an

open set of Cn and f : Ω → Cm is a continuous function. Write (z1, . . . , zn) for the

standard coordinates in Cn, and f = (f1, . . . , fm). We say that f is holomorphic

if each fj is holomorphic on each variable zk, for j = 1, . . . , m and k = 1, . . . , n.

If each fj has partial derivatives, then f is holomorphic if and only if it satisfies

“individual” Cauchy-Riemann equations, i.e. ∂fj/∂zk = 0.

Let Y be a 2n dimensional manifold; that is, every point of Y has a neigh-

bourhood homeomorphic to an open set of R2n. We can identify R2n with Cn in a

standard way, (x1, x2, . . . , x2n−1, x2n) 7→ (x1 + ix2, . . . , x2n−1 + ix2n); it makes sense

then to talk about holomorphic changes of coordinates and complex structures, in

a way parallel to the definitions of §3. A complex manifold is a (topological)

manifold with a complex structure. It is natural to say that Y has complex dimen-

sion n (and real dimension 2n). From this approach, a Riemann surface is just a

complex manifold of (complex) dimension 1 (or in another terminology, a complex

curve). We leave it to the reader to define holomorphic mappings between complex

manifolds and study their basic properties.

2.4.2. Definition. A holomorphic vector bundle of rank n on a Rie-

mann surface X is a complex manifold V , together with a surjective holomorphic

mapping π : V → X, such that for all p ∈ X, the fibre Vp = π−1(p) has the structure

of a complex vector space of (complex) dimension n, and π is locally trivial in the

following sense:

(1) for every point p ∈ X there exists a neighbourhood U , and a biholomorphic


mapping ψU : π−1(U) → U × Cn, such that the following diagram is commutative:

ψ−1(U)ψU

//

π##FF

FFFF

FFF

U × Cn

p1xx

xxxx

xxxx

U

Here p1 is the projection in the first factor, p1(p, w) = p.

(2) Let p, U and ψU be as in (1). Define a mapping ψp : Vp → Cn by ψU (v) =

ψU(p, ψp(v)). Then ψp is linear isomorphism of vector spaces.

2.4.3. A trivial example of a bundle is given by the product X×Cn → X with

the projection on the first coordinate. The isomorphism ψX is given by the identity

function. Any bundle is locally like the trivial bundle (condition (1) in the above

definition). So in some sense a bundle consists of many copies of the trivial bundle,

defined on open subsets of X and “glued” together by certain transition functions

(see 2.4.6 below) in the same way that a Riemann surface is a collection of open sets

of C related by holomorphic changes of coordinates.

2.4.4. Observe that the integer n in definition 2.4.2 is the same for all the

fibres. The complex dimension of the manifold V is n + 1. A bundle of rank 1 is

called a line bundle.

By an abuse of notation some times we will say that V is a (vector) bundle on X

(although the projection π and the mappings ψU are part of the definition).

Since our interest is in Complex Analysis, we have used holomorphic functions and

complex manifolds in 2.4.2. One can define in a similar way continuous or smooth

bundles. All bundles in this book are assumed to be holomorphic.

2.4.5. The functions ψU are called local trivializations of the bundle. From

condition (1) we get that these functions are of the form

(3) ψU(p) = (π(p), φU(p)), p ∈ π−1(U),

where φU : π−1(U) → Cn.

The Picard Group


2.4.6. There is a natural correspondence between bundles and certain cocycles

on X that we explain next for the case of line bundles, since this is the setting we

are interested in. Let Lπ→ X be a line bundle and U = Ujj∈J an open covering

of X; assume there are local trivializations ψj = ψUj: π−1(Uj) → Uj × C defined

on the sets Uj ’s. If Uj ∩ Uk 6= ∅ the mapping ψk ψ−1j , defined on (Uj ∩ Uk) × C,

should be the identity in the first variable (by (1) in 2.4.2 or (3) above) and a linear

isomorphism in the second; that is,

ψk ψ−1j : (Uj ∩ Uk) × C → (Uj ∩ Uk) × C

(p, v) 7→ (p, Tp(v)).

Since Tp is an isomorphism of 1-dimensional vector spaces it must be of the form

v 7→ λp v, where λp is a non-zero complex number. We then obtain holomorphic

functions, gjk : Uj ∩Uk → C∗, defined by gjk(p) = λp. These functions are called the

transition functions of L. This procedure associates to a line bundle L an element

(gjk) of the group of cochains C0(U ,O∗). For convenience we will use multiplicative

notation for this sheaf; in this way, a cocycle satisfies fjkfklflj ≡ 1. With this

notation in mind, the cochain (gjk) is actually a cocycle, since

(x, w) = (φj φ−1k φk φ−1

l φl φ−1j )(x, w) = (x, gjk(x) gkl(x) glj(x)w).

So we have an assignment L (gjk) of a element of Z1(U ,O∗) to each line bundle

on X, which has a given trivialization on each Uj ∈ U .

2.4.7. For the converse construction we start with a cocycle (gjk), and take

the disjoint union L1 =⊔j Uj × C. We define an equivalence relation ∼ in L1 as

follows: for (p, v) ∈ Uj × C and (q, w) ∈ Uk × C we set (p, v) ∼ (q, w) if p = q and

gjk(v) = w. The quotient space L = L1/ ∼ is a line bundle on X with transition

functions (gjk) (exercise 47).

2.4.8. These two constructions give us a correspondence between line bundles

and cocycles (with coefficients in O∗) on X. However one would like to actually

have a relation involving the cohomology group H1(X,O∗) rather than groups of

cocycles. For that purpose we need to consider isomorphism classes of line bundles.

Definition. A holomorphic mapping F : V →W between two vector bundles,


π1 : V → X and π2 : W → X on a Riemann surface X, is called a (bundle)

morphism if π2 F = π1, and F |π1−1(p) : π−1

1 (p) → π−12 (p) is a linear mapping, for

all p ∈X. A morphism is an isomorphism if there exists a morphism F−1 : W → V ,

such that F F−1 = IdW and F−1 F = IdV .

Two bundles are isomorphic if there exists an isomorphism between them.

2.4.9. Definition. A (holomorphic) vector bundle Eπ→ X of rank n is called

trivial if it is isomorphic to the bundle X × Cn p1→ X.

2.4.10. As explained above, our aim is to identify isomorphism classes of line

bundles on X with H1(X,O∗). Since this last set is a group, the space of (isomor-

phism classes of) line bundles will get a group structure from such an identification.

However there exists an operation on line bundles that will turn out to be the “right”

one under the isomorphism we are looking for (namely, the bijection between the

set of line bundles and the cohomology group will be a group homomorphism). We

explain that operation next: let L and L be two line bundles on X, and take an

open covering U = Ujj∈J , such that there are local trivializations on Uj for both

bundles. Let (gjk) and (gjk) denote the transition functions (with respect U) of L

and L, respectively. The product (gjkgjk) is an element of Z0(U ,O∗) (recall that we

are using multiplicative notation for the sheaf O∗) and thus it defines a line bundle

on X, called the tensor product of L and L, denoted by L⊗ L.

The identity element (in the set of line bundles) is given by the bundle correspond-

ing to the trivial cocycle; that is, hjk ≡ 1. An easy computation shows that the

bundle constructed from this cocycle is just the trivial bundle X × Cp1→ X (with

the notation of 2.4.7, the relation in L1 is the trivial one).

2.4.11. The inverse element of L will be the line bundle defined by the cocycle

(1/gjk). Observe that these are well defined functions because gjk takes values in

C∗. The bundle constructed from (1/gjk) is called the dual bundle of L, written

as L−1.

Proposition. L⊗ L−1 is trivial.

Proof. The proof is clear from the correspondence between line bundles and

cocycles.


2.4.12. A right inverse of the projection π : L → X is a holomorphic way of

assigning to every point of X an element of L lying on the fibre of that point.

Definition. A (holomorphic) section of a line bundle Lπ→ X is a holomorphic

mapping s : X → L such that π s = IdX.

If U = Uj of X is a covering with local trivializations ψj and s a section of L,

we have (ψj s)(p) = (p, fj(p)), where fj : Uj → C are holomorphic functions. It

is easy to see that fj = gjkfk. Conversely, any set of functions fj satisfying these

identities defines a section of L→ X in the obvious way.

If L and L are two line bundles, with sections s and s, given by (fjk) and (fjk)

respectively; then fj fj induces a section of L⊗ L, denoted by s⊗ s.

By an abuse of notation we will denote by L(U) the sheaf of holomorphic sections

of the bundle L defined on the open set U , and by L the space of global sections,

i.e. L(X).

2.4.13. Since the fibres Lp (for p a point of X) are vector spaces it makes sense

to talk of the zero vector of Lp. Under a local trivialization ψU : π−1(U) → U×C the

zero vector (of Lp) is mapped to (p, 0); in other words, the zero vector is ψ−1U (p, 0).

It is easy to check that the image of this zero vector does not depend on the choice

of ψU .

Proposition. A line bundle L → X is trivial if and only if it has a section

s : X → L such that s(p) is never equal to the zero vector of Lx, for any p ∈ X.

Proof. If s is a nowhere zero section of L → X we have that the mapping

F : X × C → L, given by F (p, λ) = λ s(p), is a bundle isomorphism.

Conversely, if F : X × C → L is an isomorphism, the section defined by s(p) =

F−1(p, 1) is nowhere zero.

For the case of the trivial line bundle X × C → X, a section that never vanishes is

given by s(p) = (p, 1) (actually we can choose any non-zero complex number instead

of 1).

Remark. The above proposition generalises to the case of vector bundles of

rank greater than 1 in a natural way: a bundle of rank n, say V → X, is trivial if


and only if there exist n holomorphic sections, sj : X → V , j = 1, . . . , n, such that

s1(p), . . . , sn(p) form a basis of Vp, for all p ∈ X. In the case of line bundles this

condition is clearly the same as the previous result, since a vector in C is a basis if

and only if it is not the zero vector.

2.4.14. Let L and L be two line bundles, F : L → L an isomorphism. Using

local trivializations we obtain functions fj : Uj → C∗, satisfying gjk = (fj/fk) gjk.

This implies that the cohomology classes of the transition functions in H1(X,O∗)

are equal, i.e. [(gjk)] = [(gjk)] ∈ H1(X,O∗). So we have an identification between

this cohomology group and the Picard group Pic(X), the group of isomorphism

classes of line bundles on X.

Proposition. The mapping Pic(X) → H1(X,O∗), that sends the isomorphism

class of a line bundle to the cohomology class of the corresponding transition func-

tions, is a group isomorphism (with the tensor product as the operation on line

bundles).

Proof. The proof follows from the following two points, that are left as exercises

to the reader:

1. if L1 and L2 are two line bundles, isomorphic to L′1 and L′

2 respectively, then

L1 ⊗ L2 is isomorphic to L′1 ⊗ L′

2;

2. let L be a line bundle, and U and V are two open coverings ofX for which L has

local trivializations. Let (gjk) and (gjk) be the elements of Z1(U ,O∗) and Z1(V,O∗)constructed from these trivializations as in 2.4.6. Then (gjk) and (tildegjk) represent

the same cohomology class in H1(X,O∗).

Divisors and line bundles

2.4.15. The definition of section can be extended to include isolated singu-

larities as follows. Associated to a (holomorphic) section s there are holomorphic

functions fj : Uj → C, with respect to some covering Uj of X. If these functions

are meromorphic we say that s is a meromorphic section of L. More formally, a

meromorphic section is a mapping s : X\A→ L, where A is a discrete subset of X,


such that

(1) π s = IdX\A, and

(2) for every point p of A, there exists a local coordinate z, with z(a) = 0, and an

integer n, such that zn s extends to a holomorphic section of L in a neighbourhood

of p.

To define the divisor of a meromorphic section, set ordp(s) = ordp(fj), if p ∈ Uj ,

the order of s at p. Since the transition functions are nowhere zero, the order is

well defined, i.e. it does not depend of the function fj. The divisor of s is given

by div(s) =∑

p∈X ordp(s) p. As an example, consider the Riemann sphere with the

standard covering U1, U2. We define a transition function by g12(z) = z, and two

functions fj : Uj → C by the expressions f1(z) = 1, f2(ξ) = 1/ξ (f2(∞) = 0). The

function g12 gives a line bundle L → C, and the pair (f1, f2) induces a section s of

L. The divisor of s consists of just one point, (s) = ∞. Observe that, unlike the

case of meromorphic functions, the degree of this divisor is not 0.

2.4.16. As remarked in 2.4.12, we denote by L the vector space (sheaf) of

(global) holomorphic sections of a bundle L→ X over X.

Theorem. For a line bundle Lπ→ X on a compact Riemann surface X the

space H1(X,L) is finite dimensional.

Proof. The proof for the case of the trivial line bundle O (2.2.10) generalises

to any line bundle without any difficulty (since all line bundles are locally trivial).

2.4.17. Proposition. Let L → X be a line bundle on a compact surface.

Then there exists a meromorphic section of L which is not holomorphic.

Proof. Fix a point p ∈ X and let U be a neighbourhood of p, with a triv-

ialization mapping ψU of the bundle L. Assume further that there is a local co-

ordinate z on U satisfying z(p) = 0. For every positive integer k we define a

meromorphic section sk of L on U\p by ψU(sk(q)) = (q, z(q)−k). Let U denote

the covering of X consisting of the two open sets U1 = U and U2 = X\p. We

define functions fjk, for j, k = 1, 2, on Uj ∩ Uk by f(k)12 = sk on U1 ∩ U2 = U\p,

f(k)21 = −f (k)

12 and f(k)11 = f

(k)22 = 0. This gives us an element f (k) of Z1(U , L). Since

H1(U , L) → H1(X,L) is injective and H1(X,L) has finite dimension, say d, we have


that

c1 f(1) + · · · + cd+1 f

(d+1) ∈ B1(U , L),

for some complex numbers (c1, . . . , cd+1) 6= (0, . . . , 0). But this is equivalent to say

that there exist holomorphic sections sj of L on Uj , j = 1, 2 satisfying

c1 s1 + · · · + cd+1 sd+1 = s1 − s2, on U\p,

for sections s1 and s2 of L (on U\p). The section s2 of L on U2 = X\p extends

to a global meromorphic section of L over X; it is not difficult to see that this section

is not holomorphic.

2.4.18. Corollary. If X is a compact surface then there exists a non-

constant meromorphic function.

Proof. Let L → X be a line bundle on X. Let p1 and p2 be two different

points of X; by the above proposition we have two sections of L, say s1 and s2 (with

p = pj in the above proof), such that each sj is holomorphic on X\pj and has a

pole at pj. We have that sj 6≡ 0, so there exists a function on X, say f , satisfying

s1 = f s2. It is easy to see that f is non-constant meromorphic function, which will

in fact have a pole at p1 and a zero at p2.

Remark. To complete the above proof we need to show that a (compact) sur-

face has line bundles. But on any Riemann surface there is a natural line bundle

called the canonical line bundle (2.5.12); see also the next subsection.

2.4.19. One can associate a line bundle L(D) to a divisor D in the following

form. Assume D =∑n

j=1 cj pj. Choose coordinate patches (Uj , zj) with zj(pj) = 0

and Uj ∩ Uk = ∅, if j 6= k. Set Un+1 = X\p1, . . . , pn. We define meromorphic

functions on Uj by fj = zcj , for j = 1, . . . , n, and fn+1 ≡ 1, and construct L(D) as

the line bundle with transition functions given by the cocycle

gjk =

fj/fk, if Uj ∩ Uk 6= ∅,

1, otherwise.


The cochain (fjk) defines a section of L(D), called the canonical section and

denoted by sD. The divisor of sD is precisely D, as one can easily check.

The holomorphic sections of the bundle L(D) can be canonically identified with the

sheaf O(D) by the mapping

O(D)(U) →L(D)(U)

f 7→f sD.

Since (f) ≥ −D, the divisor of the section f sD is effective; that is, it is a holomorphic

section of L(D). For the case of the zero divisor we have that that the sheaf Oof holomorphic functions on X is isomorphic to the sheaf of sections of the trivial

bundle X×C. In the case ofX being a compact surface this statement can be proved

directly: a section of the trivial bundle should be given by a function s : X → X×C

of the form s(p) = (p, f(p)), where f : X → C is holomorphic. So f will be a

constant function, say f(p) = λs, and the isomorphism between sections of X × C

and O is given by s 7→ λs.

It is clear from the above constructions that L(D)∗ is isomorphic to L(−D).

2.4.20. Theorem. Any line bundle is isomorphic to the line bundle of a

divisor.

Proof. The theorem follows from 2.4.17. In fact, if s is a non-zero meromorphic

section of L, with div(s) = D, then the section s ⊗ s−D is a nowhere vanishing

section of L ⊗ L(−D). Here s−D denotes the standard section of L(−D). Thus

L ∼= L(−D)∗ ∼= L(D).

2.4.21. Theorem. If L → X is a line bundle on a compact surface then

H0(X,L) is finite dimensional.

Proof. From the previous result we have that L is isomorphic to the line

bundle L(D) of a divisor D. But then (2.4.19) H0(X,L) ∼= H0(X,O(D)), so from

the Riemann-Roch theorem (2.3.3) we have that this space is finite dimensional.

2.4.22. Theorem. Two divisors D1 and D2 are linearly equivalent if and only

if the line bundles L(D1) and L(D2) are isomorphic.

2.5. SERRE DUALITY 81

Proof. Let f : X → C be a meromorphic function with div(f) = D2−D1. The

points of L(D1) are of the form (p, λ sD1(p)), for λ ∈ X, in some local trivialization.

Then the mapping

(p, λ sD1(p)) 7→ (p, λ sD2(p))

defines an isomorphism between L(D1) and L(D2) (some care has to be taken for

the case of poles and zeroes of sD1; we leave the details to the reader).

Conversely, if F : L(D1) → L(D2) is an isomorphism, then f sD1 is a section

of L(D2), and thus there exists a meromorphic function f : X → C such that

F sD1 = f sD2. Clearly div(f) = D1 −D2.

Remark. The above result can be restated as saying that the Picard group is

isomorphic to the divisor class group.

2.4.23. Definition. The degree of a line bundle L is defines as deg (D),

where D is a divisor such that L ∼= L(D).

By the above results, deg (L) is well defined.

2.5. Serre Duality

In §2.3 we have proved a first version of the Riemann-Roch theorem, which relates

the dimensions of the spaces H0(X,O(D) and H1(X,O(D)). In this section we will

prove the Serre Duality theorem, which shows that H1(X,O(D)) is isomorphic to

H0(X,Ω(−D)). In this way we obtain a version of the Riemann-Roch theorem which

has only 0th cohomology groups. We will also see that there exists a line bundle

on a Riemann surface, called the canonical bundle, whose sections are nothing but

forms on the surface.

The proofs of this section are by R.R. Simha [24].

Some results on vector bundles on Riemann sur-

faces


2.5.1. Throughout this section V → X will denote a vector bundle on a com-

pact Riemann surface. Recall that we also use V for the sheaf of holomorphic

sections of V ; that is, V (U) denotes the group of holomorphic sections of the bundle

on the open set U ⊂ X.

Proposition. If X is a compact Riemann surface and V → X a vector bundle

then H1(X, V ) is finite dimensional.

Proof. The proof is similar to the case of line bundles (2.2.10). We only need

the following result: U is an open set of X, biholomorphic to a disc D (in C),

and ψU : π−1(U) → U × Cn is a local trivialization of the bundle Vπ→ X, then

the cohomology group H1(U, V ) is isomorphic to n copies of H1(D,O), that is,

H1(U, V ) ∼=⊕

n copiesH1(D,O), and hence it is trivial.

2.5.2. Corollary. There exists a non-identically zero meromorphic section

of V → X.

2.5.3. Proposition. If V → X is a vector bundle on X then there exists a

divisor D on X, a vector bundle V ′ and an exact sequence of the form:

0 → L(D) → V → V ′ → 0.

The rank of V ′ satisfies rk(V ′) = rk(V ) − 1.

Proof. Let s : X → V be a nonzero meromorphic section of V with divisor D.

The mapping L(D) → V , given by λ sD(x) 7→ λ s(x), is an embedding of L(D) into

V . Identify L(D) with its image and define V ′ = V/L(D).

2.5.4. Proposition. H0(X, V ) is finite dimensional.

Proof. From the sequence in the proposition above we get the following exact

sequence of cohomology groups

0 → H0(X,L(D)) → H0(X, V ) → H0(L, V ′).

Assume rk(V ) = 2. Then V ′ is a line bundle so H0(X,L(D)) and H0(X, V ′) are fi-

nite dimensional, and hence so if H0(X, V ). The general cases follows from induction

on the rank of V .


2.5.5. Corollary. Let L→ C be a line bundle on the Riemann sphere. Then

the restriction of L to the complex plane, L|C → C is trivial.

Proof. We will show that L|C is trivial by exhibiting a nowhere zero holomor-

phic section. From the previous result we have a section s : C → L. Consider its divi-

sor div(s) =∑m

j=1 nj pj , and define a meromorphic function f(z) =∏

pj∈C(z−pj)−nj .

Then fs is a section of L|C which never vanishes (the function f and the section s

have poles and zeros on C at the same point, but of “opposite type”, so they cancel).

2.5.6. Given a holomorphic mapping f : X → Y between compact surfaces

and a line bundle V → X, we can construct a bundle f∗(V ) on Y as follows. First

of all, observe that to construct a bundle it suffices to give its sections on open sets.

To see this remark, consider the following (easy) result.

Lemma. Let Lπ→ X be a line bundle, ψ : π−1(A) → A×C a local trivialization.

Let x ∈ A and λ ∈ C. Then there exists a section of L on A such that s(x) = (x, λ).

Proof. Define s(y) = ψ−1(y, λ).

This shows that there are local sections whose value at a point can be fixed. So

there is always a nonzero local section. Thus, as we have claimed above, if we know

local sections, say sj, with respect to some open covering of X, we can recover

the transition functions gjk from the identity sj = gjksk, and therefore the local

trivialisations. We define the bundle f∗(V ) on Y by setting f∗(V )(A) = V (f−1(A));

that is, the sections of f∗(V ) on an open set are just the sections of V on the preimage

(under f) of the set V .

To make things a little more precise we consider the case of the function f : A→ B,

given by f(z) = zn = w, where A and B are two copies of the unit disc (we do

not use the notation D for the unit disc since we want to differentiate between the

domain and the range of the function f). The bundle on A will be the trivial bundle,

denoted by OA. We have that f∗(OA) = O(A), so the sections of f∗(OA) are just

holomorphic functions on the variable z. Assume n = 2 for the sake of simplicity.

We claim that 1, z is a basis of f∗(OA)(B). This means that any section of f∗(OA)

(on B) is of the form g1(w) 1 + g2(w) z, for holomorphic functions g1(w) and g2(w)


defined on B. Let h : A→ C be a holomorphic function (i.e. a section of the trivial

bundle OA) and write h(z) =∑∞

k=0 akzk; then we have

h(z) =a0 + a1z + a2z2 + · · · = (a0 + a2z

2 + · · · ) + z(a1 + a3z2 + · · · ) =

=(a0 + a2w + · · · ) + z(a1 + a3w + · · · ) = g1(w) 1 + g2(w) z.

The rank of f∗(OA) is equal to 2. In a more general setting of a holomorphic

function f : X → Y and L → X a line bundle we can find a situation where

B ⊂ Y is (biholomorphic to) a disc and f−1(B) has more than one component. For

example, f−1(B) = A1

⊔A2, where A1 and A2 are two discs in X. The function f

might even have different degree on the sets A1 and A2, for example we could have

f(z) = z2 and f(z) = z3 respectively. To study this case, consider again the trivial

bundle over A1

⊔A2, say OA; let z1 denote the section of f∗(OA) that is equal to z

on A1 and 0 on A2; define z2 similarly. Then one sees that a basis of the holomorphic

sections of f(OA) (on B) is given by the five functions 1, z1, 1, z2, z22. From these

two examples the reader should be able to construct f∗(L) for the general case of a

line bundle L on X. The case of vector bundles of higher rank is also easy to obtain

from the above remarks.

We have that f∗(V ) has rank equal to rk(f∗(V )) = rk(V ) deg (f), where deg (f)

denotes the degree of the function f .

The definition of f∗(V ) makes sense also in the case of f : X → Y being a proper

mapping between two (arbitrary) Riemann surfaces, since in that case the degree of

f is well defined (exercise 21 and 1.3.14).

2.5.7. Proposition. For a Riemann surface X the group H2(X,S) = 0,

where S is the sheaf of smooth functions on X.

Proof. The proof is similar to the case of the 1st cohomology group but we

include it here for the sake of completeness. Let U = Ujj be an open covering of X

and choose a cocycle (fijk) ∈ Z2(U ,S). Let φjj be a partition of unity subordinate

to U . The functions φkfijk are smooth on Ui ∩Uj. We set gij =∑

k φk fijk; observe

that for any point of X this sum has only a finite number of nonzero terms. We

have (gij) ∈ C1(U ,F) and

gij − gik + gjk = fijk,


so (fijk) splits and the result follows.

As in the case of the 1st cohomology group the above result is valid for the sheaves

of smooth forms and forms of type (1, 0) and (0, 1) as well.

2.5.8. Corollary. H2(C,O) = 0.

Proof. We have an exact sequence of sheaves on C,

0 → O → S ∂→ S(0,1) → 0,

from which we get the sequence

H1(C,S(0,1)) → H2(C,O) → H2(C,S).

Since the first and third groups in this sequence are trivial we get thatH2(C,O) = 0.

2.5.9. Now we return to the setting explained above, namely f : X → Y is

a holomorphic mapping between compact surfaces (or a proper mapping between

Riemann surfaces), V → X a bundle on X and f∗(V ) the bundle on Y constructed

using f . Since f∗(V )(A) = V (f−1(A)), if U = Ujj is a covering of Y , and we pull it

back to X via f−1, f−1(U) = f−1(Uj)j, we obtain a natural map Hn(U , f∗(V )) →Hn(f−1(U), V ), for n = 0, 1, 2. For example, for n = 0 we have

H0(Y, f∗(V )) = f∗(V )(Y ) = V (f−1(Y )) = V (X) = H0(X, V ),

In more generality,

Cn(U , V ) =∏

(f∗(V )) (U0 ∩ · · ·Un) =∏

V(f−1(U0) ∩ · · · f−1(Un)

)= Cn(f−1(U), V ).

Returning to the example of 2.5.6, f : A → B given by f(z) = z2, we have that

the mapping f∗(O(A))(B) → OA(A) given by g1(w) 1 + g2(w) z 7→ h(z) is clearly

an isomorphism. A similar construction is possible for the other example, namely

f : A1

⊔A2 → B, f |A1(z) = z2 and f |A2(z) = z3.

Theorem. The mappings Hn(Y, f∗(V )) → Hn(X, V ) are isomorphisms for

n = 0, 1, 2.

Before proving this theorem we need a technical result.


2.5.10. Proposition. If Ω is an open disc of C then H2(Ω,O) = 0.

Proof. Let (fijk) be an element of Z2(U ,O), for some covering U of Ω. By 2.5.7

we can find a smooth 1-chain, (gij) ∈ C1(U ,S), such that fijk = gjk − gik + gij.

Since ∂fijk = 0 we have ∂gij = ∂gik + ∂gjk, so (∂gij) ∈ Z1(U ,S). But H1(Ω,S) = 0

(1.5.12) so there exists a 0-chain of smooth functions, (hi) satisfying ∂gij = hi− hj .

Let ψi be smooth functions with ∂ψi = hi. It is easy to check that the functions

fij = gij − ψi + ψj are holomorphic and δ(fij) = (fijk).

2.5.11. Proof of theorem 2.5.9. The case n = 0 was proved above. For

n = 1 or n = 2, we argue as follows. Choose a covering of Y , say U = Ujj∈J ,such that f∗(V ) is trivial on Uj and each Uj is mapped to a disc in C (by some

local coordinate of Y ). Denote by f−1(U) the open covering of X consisting of

f−1(Uj)j∈J . By shrinking the sets Uj we can assume that f−1(Uj) is a disjoint

union of coordinate patches, homeomorphic to discs in C, and such that the bundle

V is trivial on each component (of f−1(Uj)). From the previous proposition we have

that U and f−1(U) are Leray coverings for f∗(V ) and V respectively. It follows that

the natural mappings Hn(U , f∗(V )) → Hn(f−1(U), V ) are bijective.

Corollary. If V is a vector bundle on a compact surface X, then the coho-

mology group H2(X, V ) is trivial.

The Canonical Line Bundle

2.5.12. On can construct in a natural way a line bundle on a surface X from

local coordinates as follows. Let (Uj, zj)j be an open covering of X by coordinate

patches. On the intersections Uj ∩ Uk we have dzj = gjk dzk, where gjk = ∂zj/∂zk.

From the chain rule of composition of functions we have that (gjk) is actually a

cocycle, so it defines a line bundle on X, denoted by KX and called the canonical

line bundle of the surface. Holomorphic (meromorphic) sections are just holomor-

phic (respectively meromorphic) 1-forms on X, as one can check by writing out the

section condition with respect to the above covering.

Recall that K denotes the canonical class of X; that is, the class of the divisor of


a holomorphic form. We have also used K to denote a representative of that class.

So with this notation we have L(K) = KX .

2.5.13. Proposition. For any divisor D the sheaves O(D + K) and Ω(D)

are isomorphic.

Proof. Let ω be a non-zero form and K = div(ω). The desired isomorphism is

given by f 7→ fω.

2.5.14. If D1 and D2 are divisors it is easy to check that L(D1 +D2) is isomor-

phic to L(D1) ⊗ L(D2). Hence L(D + K) ∼= L(D) ⊗KX ; we will use this notation

in this section.

The Riemann-Roch theorem revisited

2.5.15. We define the characteristic of a bundle V → X over a compact surface

as χ(V ) = dim H0(X, V )−dim H1(X, V ) (recall that the second cohomology group

vanishes). In particular, for the trivial line bundle we have χ(O) = 1 − g. In this

notation the Riemann-Roch theorem can be written as follows.

Theorem (Riemann-Roch). χ(L(D)) = deg (D) + χ(O) = deg (D) + 1 − g.

Corollary. If X is a compact surface, p a point of X and V a bundle on X,

the group H1(X\p, V ) vanishes.

Proof. From the Riemann-Roch theorem (2.3.4) we have that there exists a

meromorphic function f : X → C whose only pole lies at the point p. Since f is

proper on X\p we have an isomorphism H1(X\p, V ) ∼= H1(C, f∗(V )). Thus we

need to consider only the case of X being the Riemann sphere and p = ∞. If V is

a line bundle the result has been proved in 2.5.8. For the general case we use the

sequence of cohomology groups induced by the sequence 2.5.3 and use induction on

the rank of V as in the proof of 2.5.4.

2.5.16. For a vector bundle of rank n we have that the transition functions

gjk are linear isomorphisms of Cn, so they can be naturally considered as mappings

from Uj ∩Uk to GL(n,C), gjk : Uj ∩Uk → GL(n,C), where GL(n,C) is the group of

invertible square matrices of order n. The determinant of this matrix of functions is


nowhere zero, det(gjk) : Uj ∩ Uk → C∗; it is easy to show that (det(gjk)) is actually

a cocycle with values in the sheaf O∗, so it defines a line bundle on X, called the

determinant line bundle of V , written as det(V ). We define the degree of V ,

deg (V ), as the degree of its determinant line bundle. To compute a section of det(V )

we need to do it only locally, so assume that we have a trivial bundle of rank n over

the unit disk, D×Cn. Let s1, . . . , sn be sections of this bundle and v1, . . . , vn a

basis of Cn. We can write each section as sj(p) = (p, sj1(p) v1 + · · ·+ sjn(p) vn). The

n-th exterior product of Cn is a one dimensional vector space with basis v1∧· · ·∧vn.It is easy to check that the exterior product of the given sections is given by

(s1 ∧ · · · ∧ sn)(p) =(p, (s11(p) v1 + · · · + s1n(p) vn) ∧ · · · ∧ (sn1(p) v1 + · · ·+ snn(p) vn)) =

=(p, det(sjk)(p) (v1 ∧ · · · ∧ vn)).

From this we have that s1 ∧ · · · ∧ sn is a section of the determinant line bundle of

V . Or equivalently det(V ) is just the nth exterior power of V .

2.5.17. Proposition. If 0 → V ′ → V → V ′′ → 0 is an exact sequence of

bundles, then

(1) deg (V ) = deg (V ′) + deg (V ′′), and

(2) χ(V ) = χ(V ′) + χ(V ′′).

Proof. The first part is an easy consequence of the fact that the determinant

bundle det(V ) is isomorphic to the bundle det(V ′)⊗det(V ′′). To prove (2) consider

the exact sequence

0 → H0(X, V ′) → H0(X, V ) → H0(X, V ′′) → H1(X, V ′) → H1(X, V ) →

→ H1(X, V ′′) → 0.

We have that the alternate sums of the dimensions of the spaces in this sequences

are equal, i.e.

dim H0(X, V ′) + dim H0(X, V ′′) + dim H1(X, V ) =

dim H0(X, V ) + dim H1(X, V ′) + dim H1(X, V ′′).

One can prove this from the fact that the if L : W → W ′ is a linear map then

dim (ker(L)) + dim (im(L)) = dim W (for finite dimensional vector spaces). The

result follows from this equality.


2.5.18. Corollary. χ(V ) = deg (V ) + rk(V )χ(O).

Proof. The statement has been proved for line bundles. If rk(V ) = 2, in the

sequence

0 → L→ V → L′ → 0

we have that L and L′ are line bundles. Applying the previous proposition we get

χ(V ) = χ(L) + χ(L′) = deg (L) + deg (L′) + 2χ(O) = deg (V ) + rk(V )χ(O).

The general cases follows by induction on the rank of V .

2.5.19. Theorem. deg (KX) = 2g − 2.

Proof. Let f : X → C be a non-constant meromorphic function of degree

d. By composing f with a Mobius transformation if necessary we can assume that

all its poles are simple; that is, bp(f) = 0 if f(p) = ∞. The exterior derivative

of f , df = fz dz + fz dz = fz dz, is a meromorphic form on X, or equivalently a

meromorphic section of KX . If f has a zero of order n at a point p then f(z) = zn

and df = n zn−1 dz = n zbp(f) dz. On the other hand, all poles of f are simple,

so f(z) = 1z

at such points (this expression is an abuse of notation; it has to be

understood with respect to some local coordinate z) and therefore df = −1z2

dz has

a double pole at p. We get that the degree of df is equal to∑

bp(f)6=0 bp(f) − 2d =

B − 2d, where d is the degree of f and B its total ramification number (1.3.16).

Hence deg (KX) = B − 2d.

For the trivial line bundle OX we have χ(OX) = 1− g. Since H1(C,O) = 0 (2.2.6),

χ(ObC) = 1, and therefore, by the above corollary,

χ(OX) = χ(f∗(OX)) = deg (f∗(OX)) + dχ(ObC) = deg (f∗(OX)) + d.

If we can prove that deg (f∗(OX)) = −B/2 we will have χ(OX) = −B2

+d. From the

Riemann-Roch theorem (2.5.15) we know that χ(OX) = 1− g, so B = 2(g− 1 + d).

Thus

deg (KX) = B − 2d = 2g − 2.

So we need to show that deg (f∗(OX)) = −B/2. In order to simplify notation,

write Y for the Riemann sphere. We define a function τ : f∗(OX) → OY in the

following manner. If U ⊂ Y is open we know that f∗(OX)(U) = OX(f−1(U)). For h


a holomorphic function on f−1(U) (that is, h is an element of f∗(OX)(U)), we have

that τ(h) ∈ OY (U) should be a holomorphic function on U ; if q ∈ U we set

τ(h)(q) =∑

p∈f−1(q)

(bp(f) + 1) h(p).

Clearly τ(h) is holomorphic and a bundle morphism. Let L be the determinant line

bundle of f∗(OX). We use τ to define a morphism

δ : L⊗ L→ OY

as follows. First of all recall that f∗(OX) is a bundle of rank d (the degree of f); let

(λ1, . . . , λd) and (µ1, . . . , µd) be elements of OX(f−1(U)), for any sufficiently small

coordinate neighborhood U in Y . Then (λ1 ∧ · · · ∧ λd) ⊗ (µ1 ∧ · · · ∧ µd) is a section

of L ⊗ L; we denote it by λµ and define δ(λµ) = det(τ(λi) τ(µj)) (that is, δ is

the determinant of the matrix whose (i, j)th entry is τ(λi) τ(µj).) To compute the

action of δ in local coordinates let q be a point of Y and assume first that f−1(q)

consists of one single point, say p (so bp(f) = d − 1). We have, in a proper choice

of local coordinates, f(z) = zd = w. We have seen that 1, z, . . . , zd−1 is a basis of

f∗(OU ) over W (f : U →W , where U and W are two copies of the unit disc). Then

δ is given by det(τ(zi+j))i,j=o,...,n. But

τ(zi+j) = zi+j (1 + ρi+j + · · ·+ (ρi+j)d−1),

where ρ is a (bp(f) + 1)-th root of unity. This is simply because the pre-images of

a point w in the unit disc under z 7→ zm are of the form z, λ z, . . . , λm−1 z, where

zm = w and λ is an m-th root of unity. Therefore

τ(zi+j) =

(bp(f) + 1) zi+j, if i+ j = 0, bp(f) + 1

0, otherwise.

Hence det(τ(zi+j)) is a constant multiple of z(i+j)(d−1) = wd−1 and δ is simply

multiplication by a non-zero multiple of wd−1. For example, if d = 3 and bp(f) = 2

we have w = z3 and

det(τ(zi+j)

)= det

3 0 0

0 0 3z3

0 3z3 0

= −27 z6 = −27w2 = −27wd−1.


If f−1 consists of more than one point the situation is a “sum” of computations like

that above, and again we have that we have that δ is multiplication by non-zero

multiple of wPbp(f), where the sum is taken over all p with f(p) = q.

If σ1 is a holomorphic section of L ⊗ L we have that σ2 = δ(σ1) is a holomor-

phic section of OY . In local coordinates we can write σ2(q) = wnq , and therefore

δ(σ1)(q) = λwPbp(f)σ1(q) = λwnq . From this we get σ1 = wnq−

Pbp(f). The degree

of OY is 0. The degree of L⊗ L is given by the sum of the numbers nq −∑bp(f),

which gives

2 deg (L) = deg (L⊗ L) =∑

nq −

∑

p∈f−1(q)

bp(f)

= deg (OY ) − B = −B.

This completes the proof of the theorem.

2.5.20. Corollary (of the proof). The arithmetic genus g′ = dim H1(X,O)

and topological genus of a compact Riemann surface are equal.

Proof. In the above proof we have shown that B = 2(g′−1+d), where g′ is the

arithmetic genus of X. If we apply the Riemann-Hurwitz formula to the function f

of the previous proof we get 2g − 2 = −2d+ B, where g is the topological genus of

X. It follows that g = g′.

2.5.21. Theorem (Riemann-Roch). If L → X is a line bundle on a compact

surface X we have

dim H0(X,L) − dim H0(X,KX ⊗ L−1) = deg (L) + 1 − g.

Proof. It suffices to show

(4) dim H0(X,L) − dim H0(X,KX ⊗ L−1) ≥ deg (L) + 1 − g.

If we prove this, then replacing L byKX⊗L−1 we get that both sides of the inequality

change sings, so we must have an equality.

From the previous formulation of the Riemann-Roch theorem we have that

(5) dim H0(X,L) ≥ deg (L) + 1 − g.


If deg (L) > deg (KX), then H0(X,KX ⊗L−1) = 0 and thus 5 is simply 4. It follows

that to prove 4 we can assume that L = L(D) and the inequality holds for L(D+p0),

where p0 is a point of X. It is clear that dim H0(X,L(D)) ≤ dim H0(X,L(D +

p0))+1; and similarly dim H0(X,KX⊗L(−D)) ≤ dim H0(X,KX⊗L(−D−p0))+1.

Thus

dim H0(X,L(D + p0)) − dim H0(X,KX ⊗ L(−D − p0)) ≤

dim H0(X,L(D)) − dim H0(X,KX ⊗ L(−D)) + 2.

Since we are assuming that 4 holds for L(D + p0), then we have that the equation

will fail for L(D) if and only if

dim H0(X,L(D + p0)) = 1 + dim H0(X,L(D))

and

dim H0(X,KX ⊗ L(−D)) = 1 + dim H0(X,KX ⊗ L(−D − p0)).

Let σ be in H0(X,L(D+ p0)) but not in H0(X,L(D)); and let η be in H0(X,KX ⊗L(−D)) but not in H0(X,KX ⊗ L(−D − p0)). Then σ ⊗ η = σ η is a meromorphic

form with one single pole at p0. But this is not possible because of the Residue

theorem (1.4.26).

2.5.22. Corollary. dim H1(X,L) = dim H0(X,KX ⊗ L−1).

2.5.23. Corollary. dim H0(X,KX) = g and dim H1(X,KX) = 1.

The Serre Duality Theorem

2.5.24. Proposition. Given two distinct points p and q on a compact Rie-

mann surface X, there exists a meromorphic differential on X whose only singular-

ities are simple poles at p and q, with residues 1 and −1 respectively.

Proof. Consider the line bundle L = KX ⊗ L(p+ q). Since deg (KX ⊗ L−1) =

−2 < 0, we have that dim H0(X,L) = g + 1 (2.5.21, 2.1.7 and 2.4.19). The space

H0(X,KX) is a g-dimensional subspace of H0(X,KX⊗L(p+q)), so we have a form,

say ω, in H0(X,KX ⊗ L(p + q)) which is not holomorphic (it does not belong to

H0(X,KX)). Because of the form of the divisor p + q and the Residues theorem


we have that ω must have at worst a simple pole at each of the points p and q.

Since the sum of the residues must be 0, resp(ω) and resq(ω) have opposite signs,

and non-zero (since ω is not holomorphic). A appropriate multiple of ω will be the

desired form.

2.5.25. Proposition. There exists a canonical isomorphism

Res : H1(X,KX) → C.

Proof. Choose a point p ∈ X and a coordinate patch (U, z) near p, with

z(p) = 0. Let U = U, X\p. Taking residues at p defines a mapping from resp :

Z1(U , KX) → C. This mapping is not identically 0; for example, resp(dz/z) = 1. If

η ∈ C0(U , KX) we have resp(ω) = 0. This is easy to see: η will be given by a pair

(η1, η2) of holomorphic forms on U and X\p respectively. Since η1 is holomorphic,

resp(η1) = 0. On the other hand, resp(η2) = 0 by the Residues theorem. Thus we

obtain a mapping on cohomology, resp : H1(U , KX) → C (by an abuse of notation

we use the same notation for both mappings). Since dim H1(U , KX) = 1 we have

that resp is actually an isomorphism.

To complete the proof of the proposition we need to show that the mapping resp

is independent of the point p (this is the meaning of “canonical” in the statement of

the result). Let q then be another point of X. First of all observe that if we shrink

the set U in the covering U above the mapping resp does not change. So we can

choose coordinate patches (U, z) and (V, w), defined in neighbourhoods of p and q

respectively, and such that z(p) = 0 = w(q) and U ∩ V = ∅. Let U be as above,

V = V, X\q and W = U, X\p, q, V . On H1(U , KX) we consider the

class of the form dz/z, defined on U\p (dz/z is a holomorphic form in U\p);we have seen that resp(dz/z) = 1; denote the class of this form by ωp. Similarly we

define ωq as the class of dw/w in H1(V, KX). These two forms give the same class

in H1(W, KX). More precisely we claim that ωp−ωq = δ(ω′pq), where ω′

pq is defined

as the class in H1(W, KX) of the chain given by by 0 in U ∩ V = ∅, ωpq on U\pand −ωpq on V \q, and ωpq is as in 2.5.24. The difference ωp − ωq on H1(W, KX)

is given by 0, dz/z and −dw/w on U ∩ V , U\p and V \q respectively. Since


the form ωpq has a simple pole with residue 1 at p we can write it as ω1 + dzz

on the

local coordinate z, where ω1 is a holomorphic form. Hence ωp − ω′pq is given by ω1

on U\p. Similarly ωq − ω′pq is holomorphic on V \q. This shows that the class

of ωp − ωq splits in H1(W, KX) (the splitting is done by the class of ω′pq).

If ξ is a class in H1(X,KX) we can represent it by λ [ωp] or λ [ωq] in H1(W, KX),

where the complex number λ has to be the same since the classes [ωp] and [ωq]

are the same. If we now consider λ [ωq] as an element of H1(U , KX) we have that

resp(λωp) = λ; similarly resq(λωq) = λ. Thus we obtain a canonical mapping

Res : H1(X,KX) → C, by taking residues at any point of X.

If U is an open covering of X and L a line bundle, there is a natural mapping from

L⊗ (KX ⊗ L−1) to KX , which induces a paring

H0(X, ) × Z1(U , KX ⊗ L−1) → Z1(U , KX).

It is not difficult to see that this mapping induces a mapping on cohomology,

H0(X, ) ×H1(U , KX ⊗ L−1) → H1(U , KX).

2.5.26. Theorem (Serre Duality). The natural mapping

ξ : H0(X,L) ×H1(X,KX ⊗ L−1) → H1(X,KX)Res−→ C

is non-degenerate.

Proof. Given an element σ ∈ H0(X,L) we need to show that if ξ(σ⊗ρ) = 0 for

all ρ ∈ H1(X,KX⊗L−1) then σ = 0. Fix p ∈ X and let (U, z) be a coordinate patch

with z(p) = 0. Consider the covering U = U,X\p. By the corollary in 2.5.15,

we have that U is a Leray covering for the sheaves (of holomorphic sections of the

line bundles) L, KX and KX ⊗ L−1. The forms zn dz, for n ∈ Z, can be considered

as elements of Z1(U , KX ⊗L−1); let us use the same notation for the corresponding

classes in H1(U , KX ⊗ L−1). By computing the residues explicitly one can see that

if ξ(σ ⊗ (zn dz)) = 0 for all n then all the coefficients in the Taylor series of σ will

vanish; that is, σ = 0.

2.5.27. In the language of cohomology groups the above theorem takes the

following useful form.

2.6. APPLICATIONS OF THE RIEMANN-ROCH THEOREM 95

Theorem (Serre Duality). Let X be a compact Riemann surface and D a divisor

on X. Then the group H0(X,Ω(−D)) is isomorphic to H1(X,O(D)) ∗.

Proof. The above result shows that dim H0(X,L) ≤ dim H1(X,KX ⊗ L−1).

Combining this with 2.5.22 we actually have an equality. The theorem follows from

2.5.13.

2.6. Applications of the Riemann-Roch Theorem

In this section we obtain some simple consequences of the Riemann-Roch the-

orem. Perhaps the most striking fact shown here is that any surface of genus 0

must be (biholomorphically equivalent to) the Riemann sphere. We also re-write

the expression in the Riemann-Roch theorem using only 0-th cohomology groups,

namely H0(X,O(D)) and H0(X,Ω(−D)).

2.6.1. We have proved (2.5.20) that the topological genus g of a compact sur-

face X is equal to its arithmetic genus, i.e. g = dim H1(X,O). Using this fact

and the the Serre Duality formula we can rewrite the Riemann-Roch theorem in the

following form.

Theorem (Riemann-Roch). Let X be a compact surface of genus g and D a

divisor on X. Then

dim H0(X,O(D)) − dim H0(X,Ω(−D)) = 1 − g + deg (D).

The sheaves associated to the zero divisor are O and Ω, that is, the sets of holomor-

phic functions and forms on the surface X. In this case the Riemann-Roch theorem

gives us the following expression:

dim H0(X,O) − dim H0(X,Ω) = 1 − g.

In other words, the space of holomorphic 1-forms on X has dimension equal to

the genus of the surface (since H0(X,O) is isomorphic to C and therefore a one-

dimensional space). In particular, there are no non-zero holomorphic forms in the

Riemann sphere, a result that can be also obtain from the fact that the degree of

the canonical class of C is −2 (2.5.19).


2.6.2. Proposition. Let D be a divisor on a compact Riemann surface X.

Then

H0(X,O(−D)) ∼= H1(X,Ω(D))∗.

Proof. Let ω be a non-zero 1-form on X and K = (ω) its divisor. We have seen

that there are sheaf isomorphisms Ω(D) ∼= O(D + K) and O(−D) ∼= Ω(−D −K).

Therefore from the Serre Duality theorem we get

H1(X,Ω(D))∗ ∼= H1(X,O(D +K))∗ ∼= H0(X,Ω(−D −K)) ∼= H0(X,O(−D)).

2.6.3. The following theorem is an easy consequence of the previous result

and 2.1.7.

Theorem. Let X be a compact surface, D a divisor on X. Then

(1) if deg (D) > 2g − 2, H1(X,O(D)) = 0;

(2) If deg (D) > 0, H1(X,Ω(D)) = 0.

Corollary. If M denotes the sheaf of meromorphic functions on a compact

Riemann surface X, then H1(X,M) = 0.

Proof (of the corollary). If U = Uaa∈A is a covering of X it is easy to

construct another covering, say V = Vknk=1, with V < U , and refinement mapping

α : 1, . . . , n → A, such that Vk is a compact subset of Uα(k) (use the compactness

of X). Let ξ be an element of H1(X,M) and choose a cocycle (fab) ∈ Z1(U ,M)

to represent it. Thus fab is a meromorphic function on Ua ∩ Ub. In particular, since

Vj ∩ Vk is a compact subset of Uα(j) ∩ Uα(k), the function fab has a finite number

of poles in Vj ∩ Vk. So we can find an effective divisor D, with deg (D) > 2g − 2,

and such that div(fα(j)α(k)) ≥ −D on Vj ∩ Vk. By the previous theorem we have

that H1(X,O(D)) = 0, so there exist functions fj ∈ O(−D)(Vj), such that fjk =

fj − fk on Vj ∩ Vk. In other words, (fjk) = 0 in H1(V,M). Since the mapping

from H1(U ,M) to H1(V,M) is injective we have H1(U ,M) = 0. It follows from

lemma 1.5.10 that H1(X,M) = 0.

2.6. APPLICATIONS OF THE RIEMANN-ROCH THEOREM 97

2.6.4. Theorem. Let D be a divisor on a compact surface X of genus g and

p a point of X. Assume deg (D) > 2g − 1. Then there exists a divisor E, linearly

equivalent to D, such that E(p) = 0.

Proof. Since deg (D − p) = deg (D) − 1 > 2g − 2, by 2.6.3 the cohomology

groups H1(X,O(D)) and H1(X,O(D − p)) vanish. Therefore dim H0(X,O(D)) =

dim H0(X,O(D − p)) + 1 and hence the inclusion mapping

H0(X,O(D − p)) → H0(X,O(D))

will not be surjective. The image of this mapping is the set of meromorphic functions

f with div(f) ≥ −D and such that ordp(f) ≥ 1 − D(p). Let f be a function in

H0(X,O(D))\H0(X,O(D − p)). Then ordp(f) = −D(p), so we can write div(f) =

E − D, where E is a divisor satisfying E(p) = 0. From this construction we have

that E is linearly equivalent to D.

2.6.5. Theorem. Let π : L → X a line bundle with deg (L) > 2g − 1 on a

compact surface X of genus g and p a point of X. Then there exists a holomorphic

section s of L such that s(p) 6= 0.

Proof. The proof consists simply on rewriting the previous result in the lan-

guage of line bundles. Since any line bundle on X is isomorphic to the line bundle

of a divisor (2.4.20) we will assume that L ∼= L(D). By the previous result we have

a divisor E with E ∼ D and E(p) = 0. The canonical section sE of L(E) satisfies

(sE) = E; thus sE(p) = 0. A section of L can be obtained using the isomorphisms

L ∼= L(D) ∼= L(E) (these isomorphisms will take non-zero vectors to non-zero vec-

tors).

See [19] for a different proof of the above result.

2.6.6. Let X be a surface of the form X = Tτ ; let π : C → Tτ denote the

canonical projection (1.3.6). The Weierstrass ℘-function is defined by

℘(z) =1

z2+

∑

λ∈Z×Z\(0,0)

(1

(z − λ)2− 1

z2

),

where the sum is taken over all numbers λ = n+mτ , with (n,m) 6= (0, 0). Since ℘ is

invariant under the elements of Gτ (℘ Sn,m = ℘, for all Sn,m ∈ Gτ ), the expression


f℘(π(z)) = ℘(z) defines a meromorphic function f℘ : Tτ → C on the torus Tτ .

The only singularities of ℘ are double poles at the points of the form n + mτ ; but

all these points are mapped to a single point of Tτ , say z0 = π(0). Thus the the

divisor D = 2 z0 (in Tτ ) satisfies dim H0(Tτ ,O(D)) ≥ 2 (the constant functions are

obviously elements of this space).

If w and w1 are local coordinates in Tτ , whose domains of definition have non-

empty intersection; we have seen that w1(p) = w(p) + n +m, for n, m ∈ Z (1.3.6).

This implies that dw defines a 1-form on Tτ without zeroes or poles. In other words,

K = 0, where K is the canonical class of Tτ . Hence O(−D) ∼= Ω(−D) (2.5.13),

and since deg (−D) > 0 we have H0(Tτ ,Ω(−D)) = 0. Applying the Riemann-Roch

theorem we get

dim H0(Tτ ,O(D)) = 1 − g + deg (D) = 2.

The functions 1, f℘ form a basis of this space.

Using similar techniques one can compute the dimensions of other spaces of forms

and functions on a surface (exercise 50).

2.6.7. Theorem. There does not exists a meromorphic function on Tτ with

only one simple pole.

Proof. Assume h : Tτ → C is a meromorphic function whose only singularity

is a single pole. Without loss of generality (see remark 1 after the proof) we can

assume that the pole of h is at the point z0 = π(0). Then the divisor of h satisfies

div(h) ≥ −z0. If D = −2z0, we have that h ∈ H0(X,O(D)). Thus we can write h

as a linear combination of the constant function 1 and f℘: h = a+ b f℘. If b = 0 we

have that h is constant. On the other hand, if b 6= 0 then h will have a double pole

at z0. This contradiction proves the theorem.

Remarks. 1. If p0 = π(z0) and p1 = π(z1) are two points in Tτ the mapping

L : Tτ → Tτ given by L(π(z)) = π(z− z0 + z1) is clearly biholomorphic and satisfies

L(p0) = p1. Thus any two points on Tτ are “equivalent”.

2. Another way of proving the above theorem is as follows. Suppose that h : Tτ → C

has one single pole as its only singularity; then h must have degree 1 and Tτ would

2.7. PROJECTIVE EMBEDDINGS 99

have genus 0 (see 2.3.5). However we have included the above proof to show an

application of the Riemann-Roch theorem.

2.6.8. The following important result becomes now a straightforward conse-

quence of 2.3.5 and 2.5.20.

Theorem. A compact Riemann surface of genus 0 is biholomorphic equivalent

to the Riemann sphere.

2.7. Projective embeddings

We have seen that the Riemann sphere can be identified with the one-dimensional

complex projective space (exercise 17). In this section we generalise that result in

the following sense: we show that any compact Riemann sphere can be embedded

in a projective space (that is, X can be “smoothly” identified with a subset of a

projective space). The dimension of that projective space depends on whether the

given surface has a meromorphic function of degree 2 or not. Those surfaces with

such functions are in some sense different; they are called hyperelliptic surfaces and

we will study more properties of them in the next sections. The fact that a surface

can be considered as a subset of a projective space allows one to use many of the

results of Algebraic Geometry to obtain properties of Riemann surfaces; we will not

pursue such approach in this book, the interested reader can find more material

in [10] or [19].

We start this section by recalling the construction of (complex) projective spaces.

Using the space of holomorphic forms on a surface X we define a mapping, called

the canonical map, from X to certain projective space; we show that in “most”

situations this mapping is an embedding. In the remaining cases, by a small change

in the definition of the canonical map we can still produce an embedding of the

surface X in a projective space. We explain the relation of these embeddings with

certain spaces of forms onX. Finally, in the last part of this section we show another

way of obtaining embeddings by using sections of line bundles.

2.7.1. Let (Cn+1)∗ denote the set Cn+1\(0, . . . , 0). Define an equivalence

relation in this set, say ∼, by identifying (z0, . . . , zn) and (w0, . . . , wn) if there exists


a non-zero complex number λ, such that zj = λwj, for j = 0, . . . , n. The quotient

space (Cn+1)∗/ ∼ is called the n-dimensional (complex) projective space, written

as Pn. We will use the notation [z0 : · · · : zn] for a point in Pn. Observe that any

point in projective space has at least one coordinate not equal to 0. We put on Pn

the quotient topology induced by the natural mapping (Cn+1)∗π→ Pn : a set U of

Pn is open if and only if π−1(U) is open in (Cn+1)∗ (or equivalently, in Cn+1). This

topology makes Pn a manifold (of real dimension 2n). But as one might expect,

projective spaces are complex manifolds (of complex dimension n). To construct

an atlas we consider the covering U0, . . . , Un, where Uj is the open set defined by

Uj = [z0 : · · · : zn]; zj 6= 0; the coordinate functions are given by the mappings

ψj : Uj →Cn

[z0 : · · · : zn] 7→(z0zj, . . . ,

zj−1

zj,zj+1

zj, . . . ,

znzj

)

Changes of coordinates are holomorphic functions. For example ψ1 ψ−10 is the

function (w1, . . . , wn) 7→ (1/w1, w2/w1, . . . , wn/w1), defined for (w1, . . . , wn) ∈ Cn,

with w1 6= 0.

2.7.2. The following proposition is an easy consequence of the Riemann-Roch

and Serre Duality theorems.

Proposition. Let X be a compact surface of genus g ≥ 1 and p a point of X.

Then there exists a holomorphic form on X that does not vanish at p.

Proof. Since the genus of X is positive there does not exist a meromorphic

function whose only singularity is a simple pole at p (see remark 2 in 2.6.7). Or

in terms of cohomology spaces, the dimension of H0(X,O(p)) is 1. The space of

holomorphic forms vanishing at p is H0(X,Ω(−p)). We can compute its dimension

using the Riemann-Roch theorem as follows:

dim H0(X,O(p)) − dim H0(X,Ω(−p)) = 2 − g.


From this equation we get that dim H0(X,Ω(−p)) = g − 1. Since the space of

holomorphic forms has dimension equal to g the result follows.

For the rest of this section we will assume that X is a compact surface of genus

g ≥ 2, unless otherwise stated.

2.7.3. Let f : X → Pn be a holomorphic function, and p a point of X. If

f(p) = [w0 : · · · : wn] we can assume without loss of generality, that w0 6= 0. Choose

a local coordinate z on X and the coordinate ψ0 described above, to obtain a map

F (x) = (F1(x), . . . , Fn(x)) : U → Cn, where U is a neighbourhood of 0 (in C). The

differential of F (or f) at p is defined as the linear mapping

dpF : C →Cn

λ 7→(λF ′1(0), . . . , λ F ′

n(0)).

Some times we will identify dpF with the vector (F ′1(0), . . . , F ′

n(0)) to simplify no-

tation. The mapping f is called an embedding if it is injective and dpF is not

identically zero for any point p of X.

2.7.4. To define an mapping of X into Pn we consider a basis ω1, . . . , ωgof H0(X,Ω). Take a local coordinate z vanishing at p and write ωj = fj dz, for

holomorphic functions fj . We set i1 : X → Pg−1 by i1(q) = [f1(q) : . . . : fg(q)], for

q in a neighbourhood of p. By 2.7.2 there exists a 1-form that does not vanish at

q, so i1(q) lies in Pg−1. It is also easy to see thati1 does not depend on the choice

of local coordinate z (if we change coordinates, the values fj(q) are multiplied by a

non-zero complex number, give by the dereivative of the change of coordinates) and

so is well-defined. The mapping i1 is called the canonical mapping. Our aim is

to find under what conditions (on X) the mapping i1 is an embedding.

For that purpose, let p be an arbitrary but fixed point of X. We change the basis

ω1, . . . , ωg (to “adjust it” to p) as follows: first of all, by 2.7.3 we can assume

that f1(p) 6= 0; we then replace f1 by 1f1(p)

f1. By an abuse of notation we denote

this new function by f1 as well. Substitute now fj by fj − fj(p)f1, and again

abuse notation by using fj for the new functions (j = 2, . . . , g). Thus we have

that the basis f1, . . . , fg satisfies f1(p) 6= 0, and f2(p) = · · · = fg(p) = 0. So


i1(p) = [1 : 0 : · · · : 0]. We can use the local coordinate ψ0 near i1(p) to compute

dp(i1). We have that

(ψ0 i1)(z) =

((f2 z−1)(w)

(f1 z−1)(w), . . . ,

(fg z−1)(w)

(f1 z−1)(w)

),

for w in a neighbourhood of 0 in C (recall that z is a local coordinate on X with

z(p) = 0). Thus di1(p) is given by

((f2 z−1)′(0)(f1 z−1)(0) − (f2 z−1)(0)(f1 z−1)′(0)

(f1 z−1)2(0), . . .

. . . ,(fg z−1)′(0)(f1 z−1)(0) − (fg z−1)(0)(f1 z−1)′(0)

(f1 z−1)2(0)

)=

=((f2 z−1)′(0), . . . , (fg z−1)′(0)

).

We see from this expression that dp(i1) gives us the order of the zeroes of ωj at the

point p (for j = 2, . . . , g).

2.7.5. We first consider under what conditions the mapping i1 is not one-to-

one. This means that there exists a point q in X, q 6= p, such that i1(q) = [1 : 0 :

· · · : 0]; that is, if a holomorphic form vanishes at p, then it also vanishes at q. In

terms of cohomology groups this statement is given by the equality H0(X,Ω(−p)) =

H0(X,Ω(−p−q)). From 2.7.2 we know that dim H0(X,Ω(−p)) = g−1. If we apply

the Riemann-Roch theorem to the divisor p+ q, we get

dim H0(X,O(p+ q)) = dim H0(X,Ω(−p− q)) + 3 − g = g − 1 + 3 − g = 2.

The space of constant functions is a one dimensional subspace of H0(X,O(p + q)).

Hence there exists a non-constant meromorphic function f in O(pq)(X). Since the

genus g ≥ 2 the function f cannot have only one simple pole as its only singularity

(2.3.5 and 1.4.26). Thus f must have a pole at each of the points p and q and they

must be simple poles (because div(f) ≥ −p− q). We obtain that f is a function of

degree 2.

2.7.6. Definition. A compact Riemann surface is called hyperelliptic if it

has a meromorphic function of degree 2.


2.7.7. The previous paragraphs show that if the canonical map i1 fails to be

injective then X is hyperelliptic. We show next that if dpi1 = (0, . . . , 0) then X

is also hyperelliptic. From the last line of 2.7.4 we have that if dpi1 ≡ 0 then any

holomorphic form that vanishes at p must do it with order at least 2. Equivalently,

H0(X,Ω(−p)) = H0(X,Ω(−2p)). We apply again the Riemann-Roch theorem, this

time to the divisor 2p:

dim H0(X,O(2p)) = dim H0(X,Ω(−2p)) + 3 − g = 2;

which means again that X is hyperelliptic. We formulate this results in a single

statement.

Theorem. If X is a compact non-hyperelliptic surface then the canonical map-

ping i1 : X → Pg−1 is an embedding.

2.7.8. To construct an embedding im : X → Pn for the case of hyperelliptic

surfaces we consider the divisor mK, where m is a positive integer and K the

canonical class of X, or rather a representative of it, given by the divisor of a

holomorphic form, say div(ω) = K. From exercise 50 we have that

dim H0(X,O(mK)) = (2m− 1)(g − 1) = d > 0.

Let f1, . . . , fd be a basis of this space; define im : X → Pd−1 by im(p) = [f1(p) :

· · · : fd(p)]. We first need to show that this mapping is well defined, i.e. the point

im(p) is actually in Pd−1. Recall that there exists a holomorphic 1-form, say ωp

that does not vanish at p. The space of holomorphic forms Ω can be identified

with H0(X,O(K)) (2.5.13). Let f be the function in this space corresponding to

ωp under the said isomorphism; then div(f) ≥ −K and f(p) 6= 0. The function fm

is in H0(X,O(mK)) and does not vanish at p. This shows that im is well defined.

Without loss of generality we can assume that im(p) = [1 : 0 : · · · : 0]; that is

f2, . . . , fq are a basis of H0(X,O(mK − p)). Assume that there exists a point q,

with q 6= p and im(q) = im(p). This means that any function in H0(X,O(mK))

which vanishes at p it will also vanish at q. Therefore

dim H0(X,O(mK − p)) = dim H0(X,O(mK − p− q)).


It is not difficult to see (see exercise 50) that the space H0(X,O(mK − p)) has

dimension d−1. If we now apply the Riemann-Roch theorem to the divisormK−p−qwe obtain

dim H0(X,O(mK − p− q)) = dim H0(X,Ω(p+ q−mK)) +m(2g− 2)− 2 + 1− g,

which implies that

dim H0(X,Ω(p+ q −mK)) = 1.

The sheaves Ω(p+ q−mK) and O(p+ q− (m− 1)K) are isomorphic (2.5.13); thus,

from 2.1.7 we have that the degree of p+ q − (m− 1)K must be non-negative. An

easy computation shows that this condition is given by (m−1)(g−1) ≤ 1. Therefore

if m ≥ 3 we have (m− 1)(g − 1) > 1, since we are assuming that g ≥ 2, and in this

case im will be one-to-one.

If dim(p) ≡ 0 we have that any function in H0(X,O(mK)) that vanishes at p it

does it with order at least 2. Equivalently,

H0(X,O(mK − p)) = H0(X,O(mK − 2p)).

Reasoning as above, with q replaced by p, we see that if m ≥ 3 then dim(p) does

not vanish.

2.7.9. We collect all the above results in the following theorem.

Theorem. Let X be a compact Riemann surface of genus g ≥ 2 and im : X →Pn the mth canonical map. If X is not hyperelliptic then i1 is an embedding and

n = g − 1. On the other hand, i3 is always an embedding (in this case n = 5g − 6).

The Riemann sphere C is isomorphic to P1, and a torus can be embedded into P2.

Proof. The statement on the Riemann sphere was left to the reader (exercise

17) and the one regarding the torus is not difficult. We give some hints in exercise

54.

2.7.10. The above study in the case of hyperelliptic surfaces can be done in a

slightly different but equivalent way as we explain next. The reader can skip this

subsection without any problem.

Similar to the definition of 1-forms one has the concept of higher order forms: a

holomorphic m-form α is an assignment of a holomorphic function f to each local


coordinate z, such that the expression f(z) dzm is invariant. More precisely, if t is

another coordinate on X (with domain of definition not disjoint from that of z) and

α is given by the function g in this coordinate (α is g(t) dtm), then f(z) = g(t) ( ∂t∂z

)m.

For example, if ω = h(z) dz is a holomorphic 1-form then ωm := hm(z) dzm defines

an m-form, as the reader can easily check.

The space of m forms can be identified with the cohomology group H0(X,mK),

where K is the canonical class (or a representative of it). To see this choose a non-

zero holomorphic 1-form, say ω, and let K = div(ω). Let Ωm(X) denote the space

of m-forms. Then the mapping H0(X,mK) → Ωm(X) given by h 7→ hωm is an

isomorphism. The facts that im is not injective, or dpim ≡ 0 can be translated to

results on higher order forms, similar to the statements regarding 1-forms and i1.

2.7.11. Holomorphic sections of (certain) line bundles produce embeddings of

Riemann surfaces into projective spaces as well. For a line bundle L on a compact

surface X we define a mapping φL : X → Pn, where n = dim H0(X,L) − 1, as

follows. Choose a basis s0, . . . , sn of the space H0(X,L) of holomorphic sections

of L. For a point p of X let σ be a section of L, defined on a neighbourhood U of p,

which does not vanish at p. If the sections sjnj=0 do not have common zeroes the

expression [ s0(q)σ(q)

: · · · : sn(q)σ(q)

] defines a point of Pn.

The bundle L is isomorphic to the line bundle L(D) of a divisor D and the space

H0(X,L) can be identified with H0(X,O(D)) (2.4.19). In the case of deg (D) >

2g − 2 we have that

dim H0(X,O(D)) = deg (D) + 1 − g,

so n = deg (D) − g.

Theorem. If L is a line bundle of degree greater than 2g on a compact surface

X, the mapping φL : X → Pn, for n = deg (L) − g is an embedding.

Proof. Let q be an arbitrary point of X (the case q = p is also allowed; as a

matter of fact, we will use it below). Since the degree of the bundle L ⊗ L(−p) is

greater than 2g − 1 we have that there exists a section s with s(q) 6= 0 (2.6.4). Let

sp denote the standard section of L(p) (so div(sp) = p). The expression s = s⊗ sp

defines a section of L⊗ L(−p) ⊗ L(p). Since this last bundle is isomorphic to L we


can consider s as a section of L. If q 6= p the section s does not vanish at q; on the

other hand, if q = p we have that s has a simple zero at p.

Using these results, an argument similar to the above constructions finishes the proof

of the theorem. We left the details for the exercises (exercise 55).

2.7.12. Definition. A line bundle L on a surface X is called ample if the

mapping φL′ defined above is an embedding, where L′ = L⊗m = L⊗ m· · · ⊗L, for

some positive integer m. The line bundle L is called very ample if m = 1 (that is,

sections of L produce an embedding of X into projective space).

Using the above results it is easy to see when a line bundle is ample.

Proposition. A line bundle L is ample if and only if deg (L) > 0.

Proof. Assume that deg (L) > 0. For m positive integer we have

deg (L⊗m) = m deg (L) > 2g,

so the sections of L⊗m give an embedding of X from the previous result (2.7.11).

On the other hand, if L is ample, and L ∼= L(D), then dim H0(X,O(mD)) > 0,

so deg (L⊗m) = m deg (L) = m deg (D) must be positive.

Theorem 2.7.11 can be reformulated as saying that if L is a line bundle of degree

greater than 2g then L is very ample.

2.7.13. The following theorem is an easy consequence of the results of this

section.

Theorem. Let X be a compact Riemann surface of genus g ≥ 2. If X is not

hyperelliptic then the canonical bundle KX is very ample.

2.8. Weierstrass Points and Hyperelliptic Surfaces

We have seen that given an arbitrary point p on a compact Riemann surface X

of genus g there exists a meromorphic function whose only singularity is a pole at

p, of order at most g + 1 (2.3.4). It is therefore of interest to know whether there

are points on a surface that are poles of functions with order strictly less than g+1.

In this section we prove that such points, called Weierstrass points, always exist

2.8. WEIERSTRASS POINTS AND HYPERELLIPTIC SURFACES 107

on a compact surface; we also study the relationship between Weierstrass points

and hyperelliptic surfaces. Throughout this section we will assume that all compact

surfaces have genus greater than 1, unless otherwise stated.

Weierstrass Points

2.8.1. We start with a theorem of Weierstrass that gives some information

about orders of poles of meromorphic functions on compact surfaces.

Theorem (Weierstrass Gap Theorem). Let X be a compact Riemann surface of

genus g ≥ 1 and p a point of X. Then there exist exactly g integers,

1 = n1 < · · · < ng < 2g,

such that there does not exist a meromorphic function on X whose only singularity

is a pole of order nj at p.

Proof. The genus 1 case has already been proved in 2.6.7. For the case of g ≥ 2,

first of all observe that if there exists a meromorphic function on X, holomorphic

on X\p and with a pole of order k at p, then f belongs to H0(X,O(kp)) but not

to H0(X,O((k − 1)p)). From the sequence

0 → O((k − 1)p) → O(kp) → C,

we obtain the following exact sequence:

0 → H0(XO((k − 1)p)) → H0(X,O(kp)) → C.

Thus dim H0(X,O(kp)) − dim H0(X,O((k − 1)p)) is either 1 or 0, depending on

whether a function f as above exists or not.

The Riemann-Roch theorem gives

dim H0(X,O(kp)) − dim H0(X,O((k − 1)p)) =

1 + dim H1(X,O(kp)) − dim H1(X,O((k − 1)p)).


From 2.6.3 we have that dim H1(X,O(kp)) = 0 if k > 2g − 2. Therefore

2g−1∑

k=1

(dim H0(X,O(kp)) − dim H0(X,O((k − 1)p))

)=

=

2g−1∑

k=1

(1 + dim H1(X,O(kp)) − dim H1(X,O((k − 1)p))

)=

= 2g − 1 − dim H1(X,O) = g − 1.

Thus there are exactly g − 1 terms in the first sum equal to 1 and g terms equal to

0. These last summands are the ones in the statement of the theorem.

2.8.2. A look into the above proof shows that there is nothing special about

choosing one point p, the important matter is the sequence of divisors p, 2p, 3p, . . ..

Thus Weierstrass’ theorem admits the following generalisation.

Theorem (Noether Gap Theorem). Let X be a compact Riemann surface of

genus g ≥ 2. Consider a sequence of divisors, D0 = 0, D1 = p1, D2 = p1 + p2,. . . ,

D2g = p1 + · · ·+ p2g on X. Then there exist precisely g integers,

1 = n1 < . . . < ng < 2g,

such that

H0(X,O(Dj)) = H0(X,O(Dj+1))

if and only if j = nk for some k = 1, . . . , g.

2.8.3. Definition. A point p of a compact surface X is called a Weierstrass

point if there exists a meromorphic function f : X → C, such that f is holomorphic

on X\p and has a pole of order at most g at p.

From the Riemann-Roch theorem we have

dim H0(X,O(gp)) − dim H0(X,Ω(−gp)) = g + 1 − g = 1.

If p is a Weierstrass point then dim H0(X,O(gp)) ≥ 2 (the constant func-

tions form a one-dimensional subspace of H0(X,O(gp))), which implies that


dim H0(X,Ω(−gp)) ≥ 1. Thus a point p is a Weierstrass point if and only if there

exists a holomorphic 1-form on X vanishing at p with order at least g.

2.8.4. In order to study Weierstrass points we need to recall some results from

Calculus. Let f1, . . . , fm be holomorphic functions defined on a domain (connected

open set) U of the complex plane. The Wronskian of f1, . . . , fm is defined by

W (f1, . . . , fm) = det

f1 f2 · · · fm

f ′1 f ′

2 · · · f ′m

· · · · · ·f

(m−1)1 f

(m−1)2 · · · f

(m−1)m

,

where f (j) denotes the j-th complex derivative of f .

Lemma. The functions f1, . . . , fm are linearly dependent over C if and only if

W (f1, . . . , fm) ≡ 0.

Proof. One implication is clear: if a function, say fj depends linearly of the

other m − 1 functions, then the j-th column in the above matrix will be a linear

combination of the other columns and the above determinant will be 0.

We will show the other implication for the case m = 2. If W (f1, f2) ≡ 0 then

f1f′2 − f ′

1f2 ≡ 0. If f ′1 ≡ 0 then either f1 ≡ 0 or f2 is constant; in either case, f1

and f2 are linearly dependent. The case of f ′2 ≡ 0 is similar. So let us assume that

neither f ′1 nor f ′

2 is identically 0. In the connected set V = z ∈ U ; f2(z) 6= 0we have that (f1(z)/f2(z))

′ = 0, so f1 = λf2, for some complex number λ. By the

Identity Principle the equality f1 = λf2 holds on U .

We leave as an exercise to the reader (exercise 58) to show that if U is a connected,

open subset of C, and A is a discrete subset of U , then U\A is connected.

2.8.5. Let KX denote the canonical bundle of X and KnX the tensor product

of KX with itself n times, KnX = KX⊗

n· · · ⊗KX .

Theorem. Let n = g(g+1)/2. Then there exists a non-zero holomorphic section

W of KnX such that the zeroes of W are precisely the Weierstrass points of X.

The proof of this theorem requires some more material, so we postpone it till a later

subsection (2.8.9).


The transition functions of the canonical bundle KX are given by dz/dw, where

z and w are two local coordinates on X. Hence the transition functions of the line

bundle KnX are (dz/dw)n; a holomorphic section of this bundle then will be given

by a collection of holomorphic functions f(z) (in the local coordinate z) satisfying

f(z) = f(w)

(dz

dw

)n.

Let ω1, . . . , ωg be a basis of H0(X,Ω) and let p be a point of X. If z and w are

two local coordinates defined on neighbourhoods of p, we have ωj = fjdz = hjdw.

Hence fj = hjφ, for φ = dw/dz, from which it follows that

dmfjdzm

= φm+1

(dmhjdwm

)+

m−1∑

k=0

ψk

(dkhjdwk

);

where ψk are holomorphic functions independent of j. For example,

dfjdz

=dhjdw

(dw

dz

)2

+ hjd2w

dz2,

and

d2fjdz2

=d2hjdw2

(dw

dz

)3

+ 3dhjdw

dw

dz

d2w

dz2+

dhjdw

d3w

dz3.

But then we see that

W (f1, . . . , fg) = det

(dmfjdzm

)j=1,...,g

m=0,...,g−1

= det

(φm+1dmhj

dzm

)j=1,...,g

m=0,...,g−1

=

=

(g−1∏

m=0

φm+1

)W (h1, . . . , hg) = φnW (h1, . . . , hg).

Thus the function W (f1, . . . , fg) gives a section of KnX , for n = g (g + 1)/2. We will

denote this section by W = W (ω1, . . . , ωg; z)dzn.

2.8.6. We call the order of a zero p of W the multiplicity of p. We claim

that the zeroes of W do not depend of the choice of the basis ω1, . . . , ωg of

H0(X,Ω). If ω1, . . . , ωg is another basis then we have ωj =∑g

k=1 cjkωk, Here

(cjk)j,k=0,...,g is the matrix of change of basis, and therefore det(djk) 6= 0. This

implies W (ω1, . . . , ωg; z)dzn = det(cjk)W (ω1, . . . , ωg)dz

n; so the zeroes of W are


independent of the choice of basis, as claimed. From this argument it is also clear

that the multiplicity of a zero is independent of the chosen basis.

2.8.7. Proposition. The number of zeroes of W , counted with multiplicities,

is equal to (g − 1)g(g + 1).

Proof. Since W is a holomorphic section of the line bundle KnX it will have as

many zeroes as the degree of the bundle. This degree is equal to n times the degree

of the canonical bundle, that is

n(2g − 2) =g (g + 1)

2(2g − 2) = (g − 1) g (g + 1).

2.8.8. Given a point p in X we say that a basis ω1, . . . , ωg is adapted to

p if ordpω1 < ordpω2 < · · · < ordpωg. We will denote ordpωj by mj . For a basis

adapted to p we define

m(p) =

g∑

j=1

(mj − j + 1).

Proposition. The multiplicity of a point is m(p).

Proof. Since the zeroes, and their multiplicity, do not depend of the basis of

holomorphic forms we can choose a basis adapted to a point p.

Although we are assuming that X has genus g ≥ 2 the construction of W makes

sense when g = 1; in that case the statement of the proposition holds clearly. The

case of higher genus follows by induction on g, as we prove next.

Observe that for holomorphic functions f, f1, . . . , fn we have

W (f f1, . . . , f fn) = fnW (f1, . . . , fn).

Assume that the proposition has been proved for g ≤ k. For the k+ 1 case we have

W (f1, . . . , fk+1) = fk+1W (1, f2/f1, . . . , fk+1/f1) =

=fk+11 W ((f2/f1)

′, . . . , (fk+1/f1)′).

Since the basis chosen is adapted to p we have mj −m1 + 1 > 0, for j ≥ 2. By the

induction hypothesis we see

ordpW (f1, . . . , fk+1) = (k + 1)m1 +

k+1∑

j=1

(mj −m1 + 1 + j − 2),


which is the statement we wanted to show.

2.8.9. Proof of 2.8.5. As we have seen the section W (ω1, . . . , ωg; z)dzn has

a zero of order m(p) =∑g

j=1(mj − j+1) to p. We have that m(p) > 0 if and only if

mg > g− 1 (since we are assuming the basis is adapted at p). But this is equivalent

to saying that p is a Weierstrass point of X.

Hyperelliptic Surfaces

2.8.10. Recall that a compact surface is called hyperelliptic if it has a mero-

morphic function of degree 2 (definition 2.7.6). Clearly the Riemann sphere and

the torus (1.3.6) are hyperelliptic. In the first case one can take, for example, a

polynomial of degree 2. For the case of the torus the function f℘, constructed from

the Weierstrass ℘-function (2.6.6), has degree 2. We show in the next result that

surfaces of genus 2 are also hyperelliptic.

Proposition. A compact Riemann surface of genus 2 is hyperelliptic.

Proof. Let D be the divisor of a holomorphic form, so deg(D) = 2. By the

Riemann-Roch theorem we have

0 ≤ dim H0(X,O(D)) = 1 + dim H0(X,Ω(−D)).

Since dim H0(X,Ω(−D)) = 1 we have that H0(X,O(D)) = 2. The divisor D is

effective, so the constant functions form a 1-dimensional subspace of H0(X,O(D)).

Let f be then a non-constant function in H0(X,O(D)). Since X has genus 2 the

funtion f cannot have degree 1 (it would be an homeomorphism between X and the

Riemann sphere). Thus f must have degree 2 and X is hyperelliptic.

2.8.11. Proposition. If X is a hyperelliptic Riemann surface and f : X → C

is a function of degree 2 then the ramification points of f are precisely the Weierstrass

points of X.

Proof. First of all, by the Riemann-Hurwitz formula (1.3.16) the function

f has 2g + 2 ramification points. Choose one of those points, say p. We have

denoted by n1, . . . , ng the integers (gaps) between 1 and 2g given by the Weierstrass

Gap Theorem. If f(p) = ∞ then p is a pole of order 2. Otherwise, the function


h(q) = 1/(f(q) − f(p)) has a double pole at p. This means that there exists a

meromorphic function, say k, whose only singularity is a double pole at p (k = f or

k = h). The powers of this function, k, k2, . . . , kg gives us a sequence of functions

with singularities given by poles of orders 2, 4, . . . , 2g at p (and no other pole). Thus

mj = nj − 1 = 2j − 2, so mg = 2g − 2 > g − 1 (since we are assuming that g > 1)

and p is a Weierstrass point of X. To complete the proof we need to show that these

are all the Weierstrass points of the surface.

From these computations we get that the multiplicity of each ramification points

of f is given by

g∑

j=1

(mj − j + 1) =

g∑

j=1

(2j − 1 − j + 1) =

g∑

j=1

j =g (g − 1)

2.

Since f has 2g + 2 ramification numbers, we obtain that these points contribute

(2g + 2)g(g − 1)

2= (g + 1) g (g − 1)

to the number of Weierstrass points (counted with multiplicities). But then,

from 2.8.5 and 2.8.7, we see that there cannot be more Weierstrass points.

2.8.12. The next result is a consequence of the proof of the previous proposi-

tion.

Proposition. There are exactly 2g+2 Weierstrass points on a compact hyper-

elliptic surface of genus g and each point has multiplicity g (g − 1)/2.

It can be show that the converse of the above result also holds. See for exam-

ple [7, III.7.3, pg, 95].

2.8.13. Proposition. Let X be a hyperelliptic Riemann surface of genus

g ≥ 2, and let f and h be two meromorphic functions on X of degree 2. Then there

exists a Mobius transformation M such that h = M f .Proof. It follows from the proof of 2.8.11 that if f−1(∞) = p1 + p2, (as a

divisor), and p0 is a Weierstrass point of X, then p1 + p2 ∼ 2p0. Thus the divisor

h−1(∞) = q1 + q2 is linearly equivalent to p1 + p2. Let k : X → C be a meromorphic

function with divisor p1 + p2 − q1 − q2. Multiplication by k is an isomorphism

between the spaces H0(X,O(p1 + p2)) and H0(X,O(q1 + q2)). With respect to the


basis h, 1 and f, 1, this isomorphism will be given by a matrix(a bc d

)with non-

zero determinant, i.e. a d − b c 6= 0. This gives us the following relations between

the functions f and h:

k h = a f + b k = c f + d.

Therefore h = Mf whereM is the Mobius transformationM(z) = (az+b)/(cz+d).

2.8.14. Using a meromorphic function f of degree 2 on a hyperelliptic surface

X we can construct a biholomorphic mapping J : X → X in the following way. If

p is a ramification point of f then we set J(p) = p. For a non-ramification point p

there exists a unique point p′ in X, p 6= p′, such that f(p) = f(p′); we set J(p) = p′.

One sees easily that J is holomorphic and satisfies J2 := J J = IdX . We call J

the hyperelliptic involution. From this construction it seems that the mapping

J depends on the choice of the function f ; however we will show (2.8.17) that this

is not the case. The quotient space Y = X/ < J >, with the quotient topology,

is a connected Hausdorff space. Since J has only a finite number of fixed points,

it is not difficult to see that Y is a Riemann surface; one can use the coordinates

of example 1.3.7 near the fixed points of J . The natural projection π : X → Y

becomes holomorphic.

2.8.15. Proposition. A compact Riemann surface X of genus g ≥ 2 is

hyperelliptic if and only if it has an involution with 2g + 2 fixed points.

Proof. We have seen one half of the proposition: if X is hyperelliptic the

involution J constructed above satisfies the conditions of the proposition.

Assume that h : X → X is an involution with 2g+ 2 fixed points. Let Y denote

the quotient space X/ < h > and π : X → Y the natural map. Then π : X → Y has

2g + 2 ramification points. Since π has degree 2, by the Riemann-Hurwitz relation

we have that the genus of Y , say g′, satisfies

2g − 2 = 2(2g′ − 2) + 2g + 2 = 4g′ + 2g − 2.

So g′ = 0, i.e., Y is the Riemann sphere and π : X → C is a meromorphic function

of degree 2. In other words, X is hyperelliptic.

2.9. JACOBIAN VARIETIES OF RIEMANN SURFACES 115

2.8.16. Corollary (of the proof). If X is a hyperelliptic compact surface

of genus greater than 1 then the fixed points of the hyperelliptic involution are the

Weierstrass points of X.

Proof. It follows from the proofs of the previous results and we leave it as an

exercise to the reader.

2.8.17. Corollary. Let X be a hyperelliptic compact surface of genus g ≥ 2.

Then the hyperelliptic involution J is the unique involution of X with 2g + 2 fixed

points.

Proof. Let f : X → C be a degree 2 function and let J : X → X be

the hyperelliptic involution constructed from f . We have that f J = f . Let

J1 : X → X be another involution with 2g + 2 fixed points. Then f J1 : X → C

is a function of degree 2, so by 2.8.13 there exist a Mobius transformation M such

that M f = f J1.

By 2.8.15 we have that the fixed points of J1 are the Weierstrass points of X.

For any of these points, say p, we have M(f(p)) = f(J1(p)) = f(p). Hence M has

2g+2 fixed points, so it must be the identity (see corollary 3.3.5). This implies that

f J1 = f ; but this is precisely the condition that defines J , so J = J1.

2.9. Jacobian Varieties of Riemann Surfaces

In 1.3.6 we saw that a surface given by C/Gτ has genus 1. In this section we will

prove the converse statement, namely any surface of genus 1 is of the form C/Gτ .

We will also show that any compact surface (of positive genus) can be embedded

into a higher dimensional torus; that is, the quotient of Cn by a “nice” group of

translations. This torus, known as the Jacobian variety of the surface, is related to

the Picard group, the group of equivalence classes of line bundles on the surface.

2.9.1. Throughout this section X will denote a compact Riemann surface of

positive genus. We know from Topology that X is homeomorphic to a sphere with

g “handles” attached (see figure 2 for a surface of genus 2). The homotopy classes

of the paths aj and bj , j = 1, . . . , g generate π1(X, x0), and the corresponding

homology classes form a set of generators of H1(X,Z). Denote by D this collection


of paths. If we cut X along D (i.e. we consider X\a1, . . . , ag, b1, . . . , bg) we get

b1

a2

a−12

a1 b−12

a−11

b−11

b2

Figure 10. Polygon ∆ corresponding to a surface of genus 2.

a surface ∆ homeomorphic to a polygon with 4g sides. Identifying the sides in the

boundary of ∆ (see figure 10 for an example of a surface of genus 2) we recover X

(as a topological manifold). Let π : ∆ → X be the natural quotient map. Since

X\D is simply connected the mapping π :∆ → X\D, is a homeomorphism. All

these facts are well known and can be found, for example, in [13].

2.9.2. Let ψ denote a smooth 1-form defined on a neighbourhood of D. We

set

Aj(ψ) =

∫

aj

ψ, Bj(ψ) =

∫

bj

ψ,

which are called the a- and b-periods of ψ, respectively.

Let z denote the identity function on the complex plane. We can define a smooth

form on ∆ as an expression of the type f(z) dz, where f is a smooth function on the

interior of ∆, and with directional derivatives in the boundary points. The standard

result of Differential Geometry apply to this setting (manifolds with boundary). If

ψ is a smooth 1-form on X we can use the mapping π : ∆ → X to get a form on ∆

as follows. For a point z0 in∆ let w0 = π(z0). Since π is a homeomorphism in

∆ we

have that w = z π−1 is a coordinate on X\D. The form ψ will be then given by

ψ = f(w) dw; it is easy to check that f(z) dz defines a form, on∆ (the lift of ψ via

π), say ψ∗, which can be extended to ∂∆. The main property of ψ∗, as one can check

using local coordinates, is that if γ is a path in ∆, then∫γψ∗ =

∫π(γ)

ψ. Details


can be found in a book on Differential Geometry. Observe that this construction

is possible also if ψ is defined only in a neighbourhood of the paths a1, . . . , bg (in

which case it yields a smmooth form in a neightborhood of ∂∆).

Let φ be a form on X; fix a point z0 in the interior of ∆, and define a function u

on ∆ by u(z) =∫ zz0φ∗, where the integral is computed over a path in ∆ joining z0

and z. Since the polygon ∆ is simply connected the function u is well defined (the

integral does not depend on the choice of path).

With this notation we have that the expression∫∂∆uψ∗ makes sense.

Lemma. ∫

∂∆

uψ∗ =

g∑

j=1

(Aj(φ)Bj(ψ) − Aj(ψ)Bj(φ)

).

Proof. Let p be a point in aj and denote by p′ the equivalent point in a−1j as

in figure 11 (that is, p and p′ project to the same point in X). Consider a path γ

with end points p and p′, like in the figure. Clearly u(p) − u(p′) =∫γφ. Since π(γ)

is homologous to b−1j we have u(p) − u(p′) = −Bj(φ). Similarly, for p in bj we get

u(p) − u(p′) = Aj(φ). By an abuse of notation we will use aj and bj for the paths

in X and the boundary curves of ∆. Then

∫

∂∆

uψ∗ =

g∑

j=1

(∫

aj

+

∫

bj

+

∫

a−1j

+

∫

b−1j

)uψ∗ =

=

g∑

j=1

∫

aj

(u(p) − u(p′))ψ∗ +

g∑

j=1

∫

bj

(u(p) − u(p′))ψ∗ =

=

g∑

j=1

−Bj(φ)

∫

aj

ψ∗ +

g∑

j=1

Aj(φ)

∫

bj

ψ∗ =

=

g∑

j=1

(Aj(φ)Bj(ψ) −Aj(ψ)Bj(φ)) .

2.9.3. Proposition. Let X be a compact Riemann surface of genus g ≥ 1

and ω ∈ H0(X,Ω). Then

Im

g∑

j=1

Aj(ω)Bj(ω) < 0.


p

p′

a−1j

aj

γ

Figure 11. Proof of lemma 2.9.2.

Proof. Let u =∫dω∗ as above. By Stokes’ Theorem

∫

∂∆

uω =

∫

∆

d(uω∗) =

∫

∆

ω∗ ∧ ω∗ =

∫

X

ω ∧ ω.

If ω is given by ω = f dz (locally) we have

ω ∧ ω = fdz ∧ fdz = |f |2dz ∧ dz = −2i|f |2dx ∧ dy.

Hence1

2i

∫

∂∆

uω = −∫

X

|f |2dx dy < 0.

Applying the previous lemma we get

0 >1

2i

∫

∆

ω ∧ ω =1

2i

g∑

j=1

(Aj(ω)Bj(ω) − Aj(ω)Bj(ω)) =

=1

2i

g∑

j=1

(Aj(ω)Bj(ω) − Aj(ω)Bj(ω)

)=

1

2i

g∑

j=1

2 i Im(Aj(ω)Bj(ω)

).

2.9.4. Corollary. Let ω ∈ H0(X,Ω). If either∫

aj

ω = 0, or Re

∫

aj

ω = Re

∫

bj

ω = 0,

for all j = 1, . . . , g, then ω ≡ 0.

Proof. The first condition is simply Aj(ω) = 0, for j = 1, . . . , g. To see that

the second condition implies that ω ≡ 0 all one needs is the identity

Im(Aj(ω)Bj(ω)) = −Re(Aj(ω)) Im(Bj(ω)) + Im(Aj(ω)) Re(Bj(ω)).


In both cases we get a contradiction with the previous theorem.

2.9.5. Corollary. Let ω1, . . . , ωg be a basis of H0(X,Ω), then the matrix

A = (Ajk) = (∫ajωk)j,k=1,...,g is invertible.

Proof. Suppose c = (c1, . . . , cg) is a vector in Cg satisfying Act = 0. Then∑g

k=1 ajkck = 0, for j = 1, . . . , g. In other words, the a-periods of the form ω =

c1ω1 + · · · + cgωg are all zero. From the previous corollary we get that ω ≡ 0, and

since the forms ω1, . . . , ωg are linearly independent we have c1 = · · · = cg = 0.

2.9.6. We will say that a basis ω1, . . . , ωg of H0(X,Ω) is normalised if A

is the identity matrix, i.e.∫ajωk = δjk. By Stokes’ theorem (1.4.22) the normalised

basis does not change if we change the paths in D by other loops in the same

homology classes. In particular we can assume that the paths (in D) avoid a fixed

point, if needed. From now onwards we will assume that we are working with a

normalised base of holomorphic forms.

Theorem (Riemann’s Bilinear Relations). Let X be a compact Riemann surface

of genus g ≥ 1. Let D be as above and ωjgj=1 a normalised basis of holomorphic

1-forms. Set Bjk =∫ajωk, j, k = 1, . . . , g, and B = (Bjk). Then the matrix B is

symmetric and has positive definite imaginary part.

Proof. Set uj =∫ pp0ωj as above. By Stokes’ Theorem (1.4.22)

∫

∂∆

ujωk =

∫

X

ωj ∧ ωk = 0.

On the other hand, by 2.9.2,

∫

∂∆

ujωk = Bj(ωk) − Bk(ωj),

which proves that B is symmetric.

To prove the second statement, let c1, . . . , cg be real numbers, not all equal to 0,

and set ω =∑g

j=1 cjωj . By 2.9.3 we have

Im

g∑

j=1

Aj(ω)Bj(ω) < 0.


Since Aj(ω) = cj are real numbers, we get

Im

g∑

j=1

Aj(ω)Bj(ω) = Im

g∑

j=1

g∑

k=1

cjckBj(ωk) =

= −g∑

j=1

g∑

k=1

cjckIm(Bjk) < 0.

2.9.7. To give a more concrete feeling of the above theorem consider the case

of X given by X = C/Gτ (with the notation of 1.3.6). One can take ∆ to be the

polygon of vertices 0, 1, τ and 1 + τ in the complex plane. Then a = a1 is given by

the path a(t) = t, 0 ≤ t ≤ 1, and b = b1 by b(t) = tτ , 0 ≤ t ≤ 1. The differential dz

is normalised since∫ 1

0dz = 1. The statement that B is symmetric is trivial, since

B is simply the complex number given by∫ τ0

dz = τ . We see that Im(τ) is positive.

2.9.8. Given two distinct points of X, say p and q, we have seen that there

exists a meromorphic form ω on X whose only singularities are simple poles at p

and q, with residues 1 and −1 respectively (2.5.24). By adding a holomorphic form

to ω we can assume that the a-periods of this latter form are all 0 (by corollary 2.9.5

the a-periods of a holomorphic form can be arbitrarily chosen). We will denote this

normalised meromorphic form by ωpq.

Theorem (Reciprocity Theorem).

∫

bj

ωpq = 2πi

∫ p

q

ωj.

Proof. By 2.9.2 we have

∫

∂∆

ujωk =

g∑

k=1

Ak(ωj)Bk(ωpq) −Ak(ωpq)Bk(ωj) = Bj(ωpq) =

∫

bj

ωpq.

Since the interior of ∆ is a simply connected set, by the Residue theorem we get

∫

∂∆

ujωpq = 2πi(uj(p) − uj(q)) = 2πi

∫ p

q

ωj .


2.9.9. On Cg consider the group Λ of translations of the form z 7→ z + λγ,

where γ ∈ H1(X,Z) and λγ = (∫γω1, . . . ,

∫γωg). The vectors λγ corresponding to γ

in D are given by (recall that the basis of H0(X,Ω) is assumed to be normalised)

λa1 = e1 = (1, 0, . . . , 0), . . . , λag= eg = (0, . . . , 0, 1),

λb1 = B1 = (B11, . . . , B1g), . . . λbg = Bg = (Bg1, . . . , Bgg),

where Bjk =∫bjωk. For any other homology class, represented by a path γ, we

have λγ = n1e1 + · · · + ngeg + m1B1 + · · · + ngBg, where nj and mj are integers.

By Riemann’s Bilinear Relations the vectors e1, . . . , eg, B1, . . . , Bg are linearly

independent over R, and Λ is a discrete free abelian group of rank 2g (Λ is isomorphic

to Z2g). Some times we will identify the transformation z 7→ z + λγ with the vector

λγ. The quotient space

J(X) := Cg/Λ

is called the Jacobian variety of X. It is the higher genus analogous of the case of

C/Gτ . It can be easily proved that J(X) is a complex manifold of dimension g (the

1-dimensional proof of 1.3.6 generalises without any difficulty). Since the elements

of Λ preserve the addition of points of Cn we have that the Jacobian variety is an

abelian group.

2.9.10. One can define a mapping, the Abel-Jacobi map A : X → J(X), by

choosing a point, called the base point of the map, say x0 ∈ X, and setting

A(x) =

(∫ x

x0

ω1, . . . ,

∫ x

x0

ωg

)mod Λ.

The notation above means that we compute the integrals over an arbitrary path

joining x0 and x and then take the corresponding class in J(X) (this is the meaning

of mod Λ in the above expression). If γ1 and γ2 are two such paths, then γ1−γ2 ∈H1(X,Z) so the point A(x) does not depend on the choice of path. We can extend

A to divisors by

A

(r∑

j=1

njxj

)=

r∑

j=1

njA(xj),

which makes sense because J(X) is a group, as we have remarked above.

Let A1 denote the Able-Jacobi mapping with another choice of base point. Then

we have that action of A1 on points of X is given by the composition of the “old”


mapping A with a translation on J(X). The mapping induced by A1 on divisors of

degree 0 is then equal to the mapping induced by A.

2.9.11. Theorem (Abel). With the above notation, a divisor D is principal

if and only if deg D = 0 and A(D) = 0 in J(X).

Proof. First of all, since meromorphic functions have as many zeroes as poles

(counted with multiplicity) the condition of deg D = 0 is natural.

Recall that if f is a meromorphic function defined on a neighbourhood of 0 in

the complex plane we can write f(z) = zng(z), with g(z) 6= 0 (in a, perhaps smaller,

neighbourhood of 0). Then

f ′(z)

f(z)=nzn−1g(z) + zng′(z)

zng(z)=n

z+g′(z)

g(z);

so f ′/f has a pole of order 1 at z = 0 with residue n.

Write D =∑r

j=1 pj −∑r

j=1 qj , where pj 6= qk, j, k = 1, . . . , r (although we may

have repetitions among the pj ’s or qj’s). Assume f is a meromorphic function on X

with div(f) = D. Since the form df/f has poles at the points pj and qj it will be

of the form:

df

f=

r∑

j=1

ωpjqj +

g∑

k=1

ckωk,

for some complex numbers ck.

If γ is a loop in X that avoids the points of D then∫γdf/f ∈ 2πiZ. This can

be easily seen by lifting to ∆ (the integral of df/f is log(f)).

Conversely, if ψ is a differential of the form ψ =∑

j ωpjqj +∑

k ckωk, satisfying

that∫γψ is in 2πiZ for all closed paths, then we can define a function f on X by

f(x) = exp(∫ xx0ψ). It is easy to check that f is well defined, and div(f) = D.

Any form ψ as above will have periods in 2πiZ if and only if all its a- and b-

periods are in 2πiZ. Thus we have reduced the statement of the theorem to the

following:

A divisor D =∑r

j=1 pj − qj is principal if and only deg D = 0 and there exist

complex numbers c1, . . . , cg, such that the form ψ =∑r

j=1 ωpjqj +∑g

k=1 ckωk has a-

and b-periods in 2πiZ.


By the normalisations made we have Aj(ψ) = cj , and by the Reciprocity Theo-

rem

Bj(ψ) =

r∑

k=1

2πi

∫ pk

qk

ωj +

g∑

k=1

ckBjk.

Thus we see that∫γψ ∈ 2πiZ if and only if there exist integers, n1, . . . , ng, and

m1, . . . , mg, satisfying

2πinj = cj,

2πimj =∑r

k=1

∫ pk

qkωj +

∑gk=1 ckBjk.

Let ω denote the vector (ω1, . . . , ωg), and∫ pqω = (

∫ pqω1, . . . ,

∫ pqωg). We have that

the above two conditions are equivalent to

r∑

k=1

∫ pk

qk

ω = −g∑

k=1

nkBk +

g∑

k=1

mkek,

where ek and Bk are as in 2.9.9. This equation is clearly equivalent to A(D) = 0

mod Λ.

2.9.12. Theorem. If X is a compact Riemann surface of genus g ≥ 1 then

the Abel-Jacobi map A : X → J(X) is an embedding.

Proof. We first show that A is injective. This is an easy consequence of the

fact that g ≥ 1. If A(p) = A(q), for two distinct points p and q of X, then D = p−qwould be a principal divisor. So we would have a meromorphic function f : X → C

with divisor div(f) = p− q. This implies that X is homeomorphic to the Riemann

sphere C (which have genus equal to 0), a contradiction.

The second point that we need to prove is that the derivative of A is not zero at

any point of X. Choose p ∈ X, and a local coordinate z near p, with z(p) = 0. The

elements of the normalised basis ω1, . . . , ωg can be written as ωj = fjdz, where fj

are holomorphic functions in a neighbourhood of p. Then the mapping A is given

by

A(q) =

(∫ q

p

ω1, . . .

∫ q

p

ωg

)=

(∫ q

p

f1dz, . . . ,

∫ q

p

fgdz

);

so its derivative is simply

dA(p) = (f1(0), . . . , fg(0)) .


If dA(p) = (0, . . . , 0) then ωj vanishes at p for all j = 1, . . . , g. But by 2.7.2 we

know that this cannot happen.

2.9.13. Corollary. If X is a compact Riemann surface of genus 1 then the

Abel-Jacobi mapping is biholomorphic.

Proof. J(X) is a complex manifold of (complex) dimension g, so when g = 1

we have that J(X) is a Riemann surface. The Abel-Jacobi mapping A thus become

a holomorphic mapping between two compact surfaces. Since A is not constant it

must be surjective (1.3.12); therefore it is a biholomorphic mapping (it is injective

by the previous result).

This corollary simply says that any surface of genus 1 is of the form C/Gτ .

2.9.14. In the case of higher genus, g ≥ 2, we cannot have a result like the

above corollary, since X has complex dimension 1 while J(X) has dimension g.

However, it is possible to construct a g-dimensional manifold from X, Sg(X), and

a generalisation of the Abel-Jacobi mapping A : Sg(X) → J(X) that is surjective.

To do this consider the symmetric group Sg of permutations of g letters. We have

that Sg acts on Xg = X × · · · ×X as follows: for σ ∈ Sg we define

Xg ∋ (x1, . . . , xg)σ∗7→ (xσ(1), . . . , xσ(g)) ∈ Xg.

Let Sg(X) = Xg/Sg denote the quotient space. One can consider Sg(X) simply as

the set of effective divisors of degree g in X. From the above description it is easy

to put a complex structure to Sg(X). We will not pursue further this point of view

here; the result we are interested is as follows.

Theorem (Jacobi Inversion Problem). The Abel-Jacobi map A : Sg(X) → J(X)

is surjective.

2.9.15. Before giving a proof of the above theorem we need a technical result.

If X has genus g ≥ 1, and D is an effective divisor of degree g, by the Riemann-Roch

theorem we have

dim H0(X,O(D)) = 1 + dim H0(X,Ω(−D)) ≥ 1.


We say that D is special if dim H0(X,O(D)) > 1. Observe that D is not special if

and only if the only holomorphic 1-form that vanishes at D is the zero form.

Lemma. Let D ∈ Sg(X), where X is a compact Riemann surface of genus g ≥ 1.

Write D = p1 + · · ·+pg, and let Uj, 1 ≤ j ≤ g, be a neighbourhood of pj in X. Then

there exists a divisor E = q1 + · · · + qg, with qj ∈ Uj, such that E is not special.

Moreover, one can choose E with qj 6= qk (for j 6= k).

Proof. Set D0 = 0, and Dj = p1 + · · · + pj , for 1 ≤ j ≤ g. Using an induction

argument we will show that it is possible to find a divisor E = q1 + · · · + qj , with

qk ∈ Uk, consisting of distinct points, and such that dim H0(X,Ω(−Ej)) = g − j.

When j = g we have the statement of the lemma.

If j = 0 there is nothing to prove. If j = 1, we have D1 = p1, and this case

follows from the Riemann-Roch theorem:

dim H0(X,Ω(−p1)) = dim H0(X,O(p1)) − 1 + g − 1 = g − 1.

Thus we can take q1 = p1, E1 = D1. Assume now that we have proven our statement

up to some j < g. Applying the Riemann-Roch Theorem for j + 1 we see

dim H0(X,Ω(Dj+1)) ≥ g − j − 1.

Let ψ1, . . . , ψg−j be a basis of H0(X,Ω(−Dj)). Since ψj+1 is not identically 0,

there is a point qj+1, in Uj+1, such that ψj+1(qj+1) 6= 0. We can vary qj+1 is a small

neighbourhood of pj+1, so we can take qj+1 6= qk, for 1 ≤ k ≤ j. Thus

dim H0(X,Ω(Dj+1)) ≤ g − j − 1,

and the lemma is proved.

2.9.16. Proof of the Jacobi Inversion Problem. Choose a non-special

divisor D = p1 + · · · + pg, with pj 6= pk (if j 6= k). For each point pj take a local

coordinate (Uj , zj), with zj(pj) = 0. If ω1, . . . , ωg is a normalised basis ofH0(X,Ω)

we write ωj = fjkdzk on Uk. Then the Abel-Jacobi map (in U1 × · · · × Ug) is given

by

(z1, . . . , zg) 7→ A(D) + (A1(z1, . . . , zg), . . . , Ag(z1, . . . , zg)),


where

Aj(z1, . . . , zg) =

g∑

k=1

∫ zk

0

fjk(zk)dzk.

Since (∂Aj/∂zk) (0, . . . , 0) = fjk(0), we have that the Jacobian of A at the origin is

given by

f11(0) · · · f1g(0)...

...

fg1(0) · · · fgg(0)

Because D is not special we have dim H0(X,Ω(−D)) = 0; it is easy to see that

this fact implies that the above matrix has rank g. Hence, by the Inverse Function

theorem, A : V1 × · · · × Vg → A(D) + U is a homeomorphism, where Vj can be

considered as a neighbourhood of 0 in C, and U is a neighbourhood of (0, . . . , 0) in

Cg.

Our goal is to show that A : Sg(X) → Cg (the lift of A to the universal cover of

J(X)) is surjective. Let c = (c1 . . . , cg) ∈ Cg. Then there exists an integer n, such

that A(D) + (c/n) belongs to A(D) + U . Thus we have an effective divisor D′ of

degree g, such that A(D′) = A(D) + (c/n). Or equivalently, n(A(D′) − A(D)) = c.

Set D1 = −nD+ nD′ + gp0, where p0 is the base point chosen to define the map A.

By the Riemann-Roch theorem,

dim H0(X,OD1) = 1 + dim H0(X,Ω−D1) ≥ 1.

Let f : X → C be a meromorphic function with (f) ≥ −D1. Write (f) = E −D1.

Then E is an effective divisor of degree g. By Abel’s Theorem

0 = A((f)) = A(E) − n(A(D′) −A(D)) = A(E) − c,

since A(p0) = 0. This completes the proof of the theorem.

CHAPTER 3

Uniformization of Riemann surfaces

3.1 The Dirichlet Problem on Riemann surfaces 128

3.2 Uniformization of simply connected Riemann surfaces 141

3.3 Uniformization of Riemann surfaces and Kleinian groups 148

3.4 Hyperbolic Geometry, Fuchsian Groups and Hurwitz’s Theorem 162

3.5 Moduli of Riemann surfaces 178

127

128 3. UNIFORMIZATION OF RIEMANN SURFACES

One of the most important results in the area of Riemann surfaces is the Uni-

formization theorem, which classifies all simply connected surfaces up to biholomor-

phisms. In this chapter, after a technical section on the Dirichlet problem (solutions

of equations involving the Laplacian operator), we prove that theorem. It turns out

that there are very few simply connected surfaces: the Riemann sphere, the complex

plane and the unit disc. We use this result in 3.2 to give a general formulation of

the Uniformization theorem and obtain some consequences, like the classification of

all surfaces with abelian fundamental group. We will see that most surfaces have

the unit disc as their universal covering space, these surfaces are the object of our

study in §§3.3 and 3.5; we cover some basic properties of the Riemaniann geometry,

automorphisms, Kleinian groups and the problem of moduli.

3.1. The Dirichlet Problem on Riemann surfaces

In this section we recall some result from Complex Analysis that some readers

might not be familiar with. More precisely, we solve the Dirichlet problem; that is,

to find a harmonic function on a domain with given boundary values. This will be

used in the next section when we classify all simply connected Riemann surfaces.

Harmonic Functions and the Dirichlet Problem

3.1.1. Recall that a real-valued function u : U → R, with continuous second

partial derivatives, is called harmonic if ∆u = ∂2u∂x2 + ∂2u

∂y2= 0.

Lemma. Let U be an open subset of the complex plane and F : U → C a

holomorphic function. Then Re(F) and Im(F), the real and imaginary parts of F ,

are harmonic functions.

Proof. Write F = u+ iv, where u and v are the real and imaginary parts of F

respectively. The Cauchy-Riemann equations says that ux = vy and uy = −vx. So

we have

∆u = uxx + uyy = (vy)x + (−vx)y = vyx − vxy = 0,

3.1. THE DIRICHLET PROBLEM ON RIEMANN SURFACES 129

since the second partial derivatives of u commute.

In 3.1.3 we will show a local converse of this result: a harmonic function is locally

the real part of a holomorphic function.

3.1.2. Let U be an open subset of C and f : ∂U → C a continuous function

defined on the boundary of U . The Dirichlet problem with data U and f consists

on finding a continuous function u : U → R, harmonic on U and such that u = f

on ∂U . As one might expect not every problem has a solution; we give one example

in 3.1.23. The next result shows that we can always find a (unique) solution for the

Dirichlet problem when the domain U is a disc.

For a complex number z0 and a positive real number r, we denote by Dr(z0) the

open disc of radius r and centre z0, and by Dr(z0) the closed disc. We will write Dr

for Dr(0).

Theorem. Let R a positive number and f : ∂DR → R a continuous function.

Set

u(z) =

1

2π

∫ 2π

0

R2 − |z|2|Reiθ − z|2 f(Reiθ) dθ , for |z| < R,

f(z) , for |z| = R.

Then u solves the Dirichlet problem with data DR and f .

Proof. For z and ξ complex numbers the function

P (z, ξ) =|ξ|2 − |z|2|ξ − z|2

is the real part of the function

F (z, ξ) =ξ + z

ξ − z,

which is holomorphic for z 6= ξ. The expression for u in DR can be rewritten as

follows:

u(z) =1

2π

∫ 2π

0

P (z, Reiθ)f(Reiθ) dθ = Re

(1

2π

∫ 2π

0

F (z, Reiθ)f(Reiθ) dθ

)=

= Re

(1

2πi

∫

|ξ|=RF (z, ξ)f(ξ)

1

ξdξ

).

If z is in DR then F is holomorphic since |z| < |ξ| = R. Hence u is the real part of

a holomorphic function and therefore harmonic.


Clearly u is continuous in DR. To complete the proof of the theorem we need

to show that u is continuous on the boundary of DR. Let G(z, ξ) = F (z, ξ)/ξ. We

have

1

2π

∫ 2π

0

P (z, Reiθ) dθ = Re

(1

2πi

∫

|ξ|=R

ξ + z

ξ − z

1

ξdξ

)=∑

|ξ|<RresξG(z, ξ) = 1.

The function G is considered as a function of ξ, where z is a fixed point of DR. If

z = 0 then G(ξ, 0) = 1/ξ, so G has only one pole at ξ = 0 with residue 1. On the

other hand, if z 6= 0 we have that G(ξ, z) = ξ+zξ (ξ−z) ; in this case, G has two poles, at

0 and z, with residues −1 and 2 respectively. We see that the sum of the residues

of G is equal to 1.

Let ξ0 be a point in ∂DR, and ǫ > 0. Since f is continuous there exists a positive

number M , such that |f(ξ)| ≤M , for all ξ ∈ ∂U . For z ∈ DR we have

u(z) − u(ξ0) = u(z) − f(ξ0) =1

2π

∫ 2π

0

P (z, Reiθ)(f(Reiθ) − f(ξ0)

)dθ.

By the continuity of f at ξ0 there exists a δ0 > 0, such that |f(ξ) − f(ξ0)| < ǫ, if

ξ ∈ ∂U satisfies |ξ − ξ0| < δ0. We partition the boundary of the disc DR into two

disjoint sets, A and B, where

A = θ ∈ [0, 2π]; |Reiθ − ξ0| < δ0,

and B = [0, 2π]\A. The set A consists of the “angles” that are close to the point ξ0

and B is its complement in the unit circle. We have

|u(z) − f(ξ0)| ≤∣∣∣∣

1

2π

∫

A

P (z, ξ)(f(ξ) − f(ξ0))dξ

∣∣∣∣+

+

∣∣∣∣1

2π

∫

B

P (z, ξ)(f(ξ) − f(ξ0)) dξ

∣∣∣∣ ≤ ǫ+M

π

∫

B

P (z, Reiθ) dθ.

The number ǫ in the above inequality comes from the fact that f(ξ)− f(ξ0) is small

for “points in” A and the total integral of P over the boundary of DR is equal

to 1. The bound of the second integral comes from the bound M of |f | and the

fact that P (z, ξ) > 0, for |ξ| > |z|. Let now z be in DR and close to ξ0; that is,

|ξ0 − z| < δ ≤ δ0/2. For θ ∈ B we have that

|Reiθ − z| ≥ |Reiθ − ξ0| − |ξ0 − z| ≥ δ02.


On the other hand,

R− |z| = |ξ0| − |z| ≤ |ξ0 − z| < δ.

Using these inequalities we have

P (z, Reiθ) =R2 − |z|2|Reiθ − z|2 ≤ (R + |z|) (R− |z|)

(δ0/2)2≤ 8Rδ

δ20

.

We can now bound the above integral as follows:

M

π

∫

B

P (z, Reiθ) dθ =M

π

8Rδ

δ20

2π =16MδR

δ20

.

For ǫ given the value of δ0 is fixed, so we can make δ small enough such that for we

have |u(z)− u(ξ0)| ≤ 2ǫ, for z as above. This shows that u is continuous at ξ0.

The function P is called the Poisson kernel.

3.1.3. In the above proof we have shown the following result.

Corollary. If u is harmonic then u is locally the real part of a holomorphic

function.

3.1.4. Corollary. Let u : DR → R be a harmonic function. Then u satisfies

u(0) =1

2π

∫ 2π

0

u(reiθ)dθ,

for 0 < r < R.

Proof. Apply the above theorem on Dr for the boundary values given by

u : ∂Dr → R and observe that P (z, 0) = 1.

3.1.5. Corollary (Mean Value Property). Let u : DR → R be a harmonic

function. Let z0 be a point in DR and r > 0 a positive number such that Dr(z0) is

contained in DR. Then

u(z0) =1

2π

∫ 2π

0

u(z0 + reiθ) dθ,

for 0 < r < R.


3.1.6. From corollary 3.1.3 one expects that harmonic functions share some of

the properties of holomorphic functions. In that sense one can consider the Mean

Value Property as the analogy of Cauchy’s Integral Formula. In the next result

we see that the Maximum Modulus Principle (1.1.9) is also satisfied by harmonic

functions.

Proposition (Maximum Modulus Principle). Let u : DR → R be a harmonic

function. If there exists a point z0 ∈ DR, such that u(z) ≤ u(z0) for all z ∈ DR,

then u is constant.

Proof. The set

E = z ∈ DR; u(z) = u(z0) = u−1(u(z0))

is closed since u is a continuous function. Let z be an arbitrary point of E and r > 0

such Dr(z) is contained in DR. From the Mean Value Property we get

u(z) =1

2π

∫ 2π

0

u(z + reiθ) dθ ≤ 1

2π

∫ 2π

0

u(z0) dθ = u(z0) = u(z).

This implies that u(z+reiθ) = u(z0) for all θ in [0, 2π]. Thus E is an open set. Since

DR is connected and E is not empty we have E = DR and therefore u is constant

on U .

A similar result with minimum instead of maximum can be obtained from the fact

that if u is a harmonic function then −u is also harmonic; we leave the details for

the reader.

3.1.7. Corollary. If u : DR → R is harmonic on DR and continuous on

DR then its maximum value is attained in the boundary of DR; that is, there exists

a point z0 ∈ ∂DR, such that u(z) ≤ u(z0), for all z ∈ DR.

Proof. Since DR is compact and u continuous there is a value u(z1) where u

attains its maximum. If z1 is in ∂DR there is nothing to prove. On the other hand, if

z1 ∈ DR, from the previous corollary we have that u is constant in DR and therefore

in DR. In this case we can choose any point of ∂DR as z0.

3.1.8. Corollary. If the Dirichlet Problem has a solution on a bounded

domain then the solution is unique.


Proof. Apply the Maximum Modulus Principle to u1−u2 and u2−u1, where u1

and u2 are two solutions of the (same) Dirichlet problem. Here the condition of

the domain being bounded is necessary. Consider for example the Dirichlet problem

on the upper half plane with boundary values given by the identically 0 function (on

the real line). Then the contant function 0 and the function Im(z) are two distinct

solutions of this Dirichelt problem.

3.1.9. Using this result we can show that the Mean Value Property (3.1.5) is

also a sufficient condition for harmonicity.

Proposition. Let u : U → R be a continuous function on an open set U of the

complex plane. Assume u satisfies the Mean Value Property, namely

u(z0) =1

2π

∫ 2π

0

u(z0 + reiθ) dθ,

for all z0 ∈ U and all positive r such that the closed disc of centre z0 and radius r,

is contained in U . Then u is harmonic.

Proof. Let v be the solution of the Dirichlet problem on Dr(z0) with values

given by the function u. Observe that the proof of the Maximum Modulus Principle

uses only the Mean Value Property. Hence from 3.1.7 we have that v − u has its

maximum on ∂Dr(z0); that is v(z)−u(z) ≤ 0 for all z ∈ Dr(z0). Applying the same

argument to the function u − v we obtain that u(z) − v(z) ≤ 0 and this completes

the proof.

3.1.10. Another similarity between harmonic and holomorphic functions is

given by the following result.

Corollary. Let u : DR → R be a sequence of harmonic functions which con-

verges uniformly on compact subsets of DR to a (continuous) function u : DR → R.

Then u is harmonic.

Proof. For z0 in DR let r > 0 be such that Dr(z0) ⊂ DR. Then we have

u(z0) = limnun(z0) = lim

n

1

2π

∫ 2π

0

un(z0 + reiθ)dθ =1

2π

∫ 2π

0

u(z0 + reiθ)dθ.

For the last equality we have used that un converges uniformly on compact sets to

u, so in particular on the circle of centre z0 and radius r. It follows from 3.1.9 that

u is harmonic.


3.1.11. The next lemma is needed to prove Harnack’s inequality.

Lemma. Let z be a complex number with |z| = s and r a positive number

satisfying s < r. Thenr − s

r + s≤ r2 − s2

|reiθ − z|2 ≤ r + s

r − s,

for any real number θ.

Proof. To prove the left hand side inequality we use that

|reiθ − z| ≤ |reiθ| + |z| = r + s.

The other inequality follows from

|reiθ − z| ≥ |reiθ| − |z| = r − s.

These two inequalities, together with the expression r2 − s2 = (r − s)(r + s), prove

the result.

Let u : DR → R+ be a positive harmonic function and z ∈ DR a point with |z| = s.

Choose a positive real number r with s < r < R; then

u(z) =1

2π

∫ 2π

0

r2 − s2

|reiθ − z|2u(reiθ) dθ ≤ 1

2π

r + s

r − s

∫ 2π

0

u(reiθ) dθ =r + s

r − su(0).

Proposition (Harnack’s inequality). Let u : DR → R be a positive harmonic

function. Then, for all z0 ∈ DR with |z| = s, and for all positive r such that

s < r < R, one hasr − s

r + su(0) ≤ u(z) ≤ r + s

r − su(0).

Proof. The right hand side inequality was proved before the statement of the

proposition. The proof of the other inequality is similar.

3.1.12. The main application of the above inequality is the proof of the fol-

lowing theorem, which is similar to Montel’s theorem (1.1.13).

Theorem (Harnack’s Principle). Let M be a real number and un∞n=1 be a

non-decreasing sequence of harmonic functions on DR satisfying un ≤M . Then the

sequence un converges uniformly on compact subsets of DR to a harmonic function

u : DR → R.


Proof. The pointwise convergence follows from the fact that un(z) is a bounded,

non-decreasing sequence of real numbers (for fixed z). Thus to complete the proof

we only need to show that the convergence is uniform on compact subsets of DR. Let

ǫ > 0 be given; then there exists an n0 such that un(0)−um(0) ≤ ǫ, for n0 ≤ m ≤ n.

Choose a real number s such that 0 < s < r < R. We apply Harnack’s inequality

to the non-negative function un − um on the disc Ds:

un(z) − um(z) ≤ r + s

r − s(un(0) − um(0)) ≤ r + s

r − sǫ.

Since ǫ can be made arbitrarily small we get that unn converges uniformly on Ds.

But any compact subset K of DR is contained in a disc of the form Ds. Thus u is

harmonic (3.1.10).

3.1.13. The following more general result can be found in [1] (for our applica-

tions the previous version of Harnack’s Principle is enough).

Theorem. Consider a sequence of functions un(z), each defined and harmonic

in certain region Ωn. Let Ω be a region such that every point in Ω has a neigh-

bourhood in all but a finite number of the Ωn, and assume moreover that in this

neighbourhood un(z) ≤ un+1(z) for n sufficiently large. Then there are only two pos-

sibilities: either un(z) tends uniformly to ∞ on every compact subset of Ω, or un(z)

converges uniformly on compact subsets of Ω to a harmonic function u : Ω → R.

Subharmonic functions

3.1.14. Finding non-trivial harmonic functions on domains is not an easy prob-

lem. What we will do is to consider a more general class of functions, called sub-

harmonic functions, which are “close” enough to be harmonic; taking limits in this

class we obtain harmonic functions. The precise definition we need is as follows.

Definition. A continuous function u : U → R on an open set U of the complex

plane is said to be subharmonic if for every harmonic function h : U → R, and

every domain V ⊂ U , the function u+ h either is constant or has no maximum (on

V ).

Suppose V is a domain with compact closure V ⊂ U . Let h : V → R be a continuous


function, harmonic on V . If u is subharmonic (on U) then the maximum of u + h

on V is attained in the boundary of V . The proof is similar to the case of harmonic

functions (3.1.7).

It is clear from the above definition that u is subharmonic if and only if it is locally

subharmonic; that is, every point of U has a neighbourhood where u is subharmonic.

3.1.15. Let D be a disc with D ⊂ U and u : U → R a subharmonic function.

Denote by PD,u the function that is equal to u on U\D and solves the Dirichlet

problem on D with boundary values given by u|∂D.

Proposition. A continuous function u : U → R is subharmonic if and only if

u ≤ PD,u for every disc D whose closure is contained in U .

Proof. Assume first that u is subharmonic on U . For any disc D with D ⊂ U

we have that u− PD,u is equal to 0 on ∂D. Since u− PD,u is continuous on D and

u is subharmonic, either u− PD,u is identically 0 or it satisfies u− PD,u ≤ 0 on D.

To prove the converse let h : U → R be a harmonic function and V ⊂ U a

domain. Assume that u+ h has a maximum value on V , say m0. Set

C = z ∈ V ; u(z) + h(z) = m0.

This set is a closed subset of V . Let z0 be a point of C and D a disc of radius r

with centred at z0 and such that D ⊂ V . Then we have

m0 = u(z0) + h(z0) ≤ PD,u(z0) + h(z0) =

1

2π

∫ 2π

0

(u(z0 + reiθ) + h(z0 + reiθ)

)dθ ≤ m0.

It follows that u(z0 + reiθ) + h(z0 + reiθ) = m0 for all θ; that is, C is an open subset

of V . Since V is connected we have that C = V and u+ h is constant on V .

3.1.16. Corollary. Let u : U → R be a continuous function. Then u is

subharmonic if and only if for every point z0 ∈ U , and every positive number r such

that Dr(z0) is contained in U , the following inequality holds:

u(z0) ≤1

2π

∫ 2π

0

u(z0 + reiθ) dθ.


Proof. If D = Dr(z0) then the right hand side of the above inequality is simply

PD,u(z0).

Corollary (Maximum Modulus Principle). Subharmonic functions satisfy the

Maximum Modulus Principle.

Proof. The proof is similar to the case of harmonic functions.

3.1.17. Proposition. Let u, v : U → R be subharmonic functions, c a posi-

tive real number, and D ⊂ U a disc. Then the functions cu, u + v, max(u, v) and

PD,u are subharmonic (on U).

Proof. The fact that cu and u + v are subharmonic follows from the above

corollary.

To show that the maximum of two subharmonic functions is subharmonic consider

a point z0 of U and assume that max(u, v)(z0) = u(z0); then we have

max(u, v)(z0) = u(z0) ≤1

2π

∫ 2π

0

u(z0 + reiθ)dθ ≤ 1

2π

∫ 2π

0

max(u, v)(z0 + reiθ)dθ.

It follows from the previous result that max(u, v) is subharmonic.

Consider now the function PD,u. Clearly this function is subharmonic on U\D (since

it is equal to u on this set) and on D (because it is harmonic). So we need to check

subharmonicity only at the points on the boundary of D. Let z0 be one such point;

using the inequality u ≤ PD,u (3.1.15) we have

PD,u(z0) = u(z0) ≤1

2π

∫ 2π

0

u(z0 + reiθ) dθ ≤ 1

2π

∫ 2π

0

PD,u(z0 + reiθ) dθ.

We can now apply the previous corollary.

3.1.18. For functions of class C2 (that is, with continuous partial derivatives

of second order) we have another characterisation of subharmonicity as follows.

Proposition. Let u : U → R be a C2 function. Then u is subharmonic if and

only if ∆u ≥ 0 on U .

This result is taken some times as the definition of subharmonic functions. However a

function does not need to have partial derivatives in order to satisfy definition 3.1.14.


3.1.19. Harmonic functions on Riemann surfaces were defined in 1.4.11. Since

a harmonic function u : U → R, defined on a domain of the complex plane, is locally

the real part of a holomorphic function one sees that harmonicity is preserved under

changes of coordinates (recall that to compute the partial derivatives of a function on

Riemann surface we need to take local coordinates). However, to define subharmonic

functions we need a little more of extra work. We begin with a definition.

Definition. A disc on a Riemann surface X is a domain D such that there

exists a local coordinate patch (U, z) with D ⊂ U and z(D) is a closed disc on C.

Given a disc D on X, and a continuous function u : X → R, we can define PD,u :

X → R as in 3.1.15.

Definition. A continuous function u : X → R defined on a Riemann surface

X is called subharmonic if for every disc D of X, and every harmonic function on

D satisfying u ≤ h one has that u ≡ h of u < h (on D).

It is easy to show that for the case of X = C this definition is equivalent to 3.1.14.

Proposition 3.1.15 and corollary 3.1.16 extend to Riemann surfaces with similar

proofs. Therefore we see that talking of the Dirichlet problem and solutions of it on

Riemann surfaces makes sense.

In a more invariant way we have the following definition.

Definition. Let X be a Riemann surface, V ⊂ X an open set of X and

u : V → R a real-valued function defined on V . We say that u is harmonic

(respectively subharmonic) if for any local patch (U, z) with U ∩ V 6= ∅, the

function

(u z−1) : z(U ∩ V ) → R

is harmonic (respectively, subharmonic).

Perron’s method

3.1.20. The idea of Perron’s method to find harmonic functions consists of

taking a family of subharmonic functions that satisfy certain conditions and then

show that the pointwise supremum of such a family must be harmonic.


Theorem (Perron’s method). Let U be a domain of the complex plane and

f : ∂U → R a bounded function. Denote by M the family of subharmonic functions

u : U → R satisfying lim supz→z0 u(z) ≤ f(z0), for all z ∈ ∂U . Then the supremum

of the family M is a harmonic function.

Proof. First of all, observe that if |f(z)| ≤ K for z in ∂U then v(z) ≤ K for

all z in U and all v in M (this is simply a consequence of the Maximum Modulus

Principle for subharmonic functions (3.1.16)).

It is easy to see that the family M has the following properties:

1. If u1 and u2 belong to M so does max(u1, u2).

2. If u ∈ M, and D is a disc contained in U then PD,u is in M.

Fix a point z0 ∈ U and let D be a disc containing z0 and satisfying D ⊂ U .

Then there exists a sequence of functions unn in M (the sequence may depend

on the point z0) such that lim supn→∞ u(z)n = u(z0). Let us define functions vn by

vn = max(u1, . . . , un). The sequence vnn is clearly non-decreasing and contained

in M (because of property 1 above). If we set wn = PD,vnwe have that wn belongs

to M (property 2). Moreover the following inequalities hold:

un(z0) ≤ vn(z0) ≤ wn(z0) ≤ u(z0).

So limn→∞wn(z0) = un(z0). Let w be the limit of the sequence wnn. Then w is

harmonic on D by Harnack’s principle and w ≤ u with w(z0) = u(z0).

Consider now another point of U , say z1, and let let u′n be a sequence and D′ a

disc similar to the ones considered above. We set u′n = max(un, u′n) and repeat the

above process to obtain a function w′ satisfying w ≤ w′ ≤ u, and w(z1) = w′(z1).

But then w(z0) = w′(z0) (since w(z0) ≤ w′(z0) ≤ u(z0) = w(z0)). So w ≡ w′ on D′.

Thus w is harmonic on the domain U .

3.1.21. Lemma. Let U be a domain in the complex plane and z0 a point of

∂U . Assume that there exists a continuous function ω : ∂U → [0,+∞) such that

ω(z0) = 0 and ω(z) > 0, for all z ∈ ∂U\z0. If f : ∂U → R is a bounded function,

continuous at z0 and M is as in theorem 3.1.20, then limz→z0 u(z) = f(z0), for

z ∈ U .

Proof. It suffices to show that


lim supz→z0 u(z) ≤ f(z0) + ǫ, and lim infz→z0 u(z) ≥ f(z0) − ǫ,

for ǫ > 0 arbitrary, z0 ∈ ∂U and z ∈ U .

Let W be a neighbourhood of z0 such that |f(z)−f(z0)| < ǫ for z ∈W . Consider

the function

g(z) = f(z0) + ǫ+ω(z)

ω0

(K − f(z0)

),

where ω0 > 0 is the minimum of the harmonic function u on U\(W ∩ U). For z in

W we have g(z) ≥ f(z0) + ǫ, while for z not in W we see that g(z) ≥ K + ǫ > f(z).

By the Maximum principle we have that if v ∈ M then v < g. Thus u ≤ g, which

implies that lim supz→z0 u(z) ≤ g(z0) ≤ f(z0) + ǫ.

The second inequality is proven in a similar way by using the function

h(z) = f(z0) − ǫ− ω(z)

ω0

(K + f(z0)

).

3.1.22. The function ω in the above lemma is called a barrier at z0. It is

clear that if every point of the boundary of U has a barrier then we can solve the

Dirichlet problem for that domain. One would like to have geometric conditions on

a domain so that we can easily see the existence of barriers at its boundary points.

An easy example is given by the upper half plane U = H = z ∈ C; Im(z) > 0.Take any point, say z0 = 0. Then ω(z) = Im(eiπ/2z) is a barrier at the origin. More

generally, let z0 ∈ ∂U and let z1 denote a point not in U . Denote by [z0, z1] the

segment joining these two points and assume that [z0, z1] ∩ U = z0. Then the

function

ω(z) = Im

(√z − z0z − z1

)

is a barrier at z0, for a proper choice of the square root.

3.1.23. We end this section with an example of a Dirichlet problem that has

no solution. Consider the open set U = D∗ = z ∈ C : 0 < |z| < 1, the punctured

unit disc, and the function f defined on ∂U = S1 ∪ 0 by f(0) = 1, f(z) = 0, for

|z| = 1. If u were a solution for the Dirichlet problem with this data, then u would

have its maximum (it must have a maximum since U is compact) at the boundary of

3.2. UNIFORMIZATION OF SIMPLY CONNECTED RIEMANN SURFACES 141

the disc (0 is an interior point of U). But this would imply that u ≡ 0, contradicting

the fact that u(0) = 1.

3.2. Uniformization of simply connected Riemann

surfaces

We have seen that a compact, simply connected Riemann surface is biholomor-

phic to the Riemann sphere. By the Riemann mapping theorem we have that nay

simply connected open subset of the Riemann sphere is biholomorphic to either

the complex plane or the unit disc. In this section we show that these three sur-

faces are the only simply connected Riemann surfaces, up to biholomorphisms. The

proof assumes only a couple of results from Complex Analysis (that we state at the

beginning) and the theory of harmonic functions; it is based on a paper of R.R.

Simha [23].

3.2.1. Theorem (Koebe). Let A be the class of one-to-one holomorphic func-

tions defined on the unit disc f : D → C and satisfying f(0) = 0, f ′(0) = 1. Then

A is normal and compact in the topology of uniform convergence on compact subsets

of the disc.

3.2.2. Theorem (Riemann Mapping Theorem). If A is a simply connected

open subset of the complex plane, with C\A not empty, then A is biholomorphic to

the unit disc.

3.2.3. Lemma. Let h : Ω → R be a harmonic function defined on an open,

connected set Ω of the complex plane. If there exists an open subset U of Ω, such

that h|U is constant, then h is constant (on Ω).

Proof. Let p0 be a fixed point of U . For any point p of Ω consider a path

γ : [0, 1] → Ω with γ(0) = p1, γ(1) = p. A harmonic function is locally the

real part of a holomorphic function (3.1.3); that is, for every point q of Ω there

exists a neighbourhood V of q, and a holomorphic function fV : V → C, such that

h = Re(fV ) on V . Since the image of γ is compact we can find connected open sets,

U0, . . . , Un, satisfying the following properties:

1. there exist holomorphic functions fj : Uj → C, such that h is the real part of


fj on Uj, for j = 0, . . . , n;

2. Uj ∩ Uj+1 6= ∅, for j = 0, . . . , n− 1.

Since U0 ∩ U is not empty, and h is constant on U , we have that the real part of f0

is constant on U0. But then f0 must be constant. Similarly we get that f1 must be

constant on U1; in particular, the real part of f1, which is equal to h|U1, is constant.

By a finite number of steps we get that h|Unis constant and therefore h(p) = h(p0);

that is, h is constant on Ω.

3.2.4. Lemma. Let f : Ω → C be a holomorphic function defined on a con-

nected open subset Ω of the complex plane. Assume that f is a (branched) covering

map of degree d onto its image f(Ω). Then

∫

Ω

f ∧ df = d

∫

f(Ω)

dz ∧ dz.

Proof. Assume first that Ω = f(Ω) = D and f is given by f(z) = zn. The proof

in this case is an easy calculation. If we write z = reiθ we have dz∧dz = −2irdr∧dθ

and therefore∫

D

df ∧ df =

∫

D

nzn−1dz ∧ dzn−1dz = −2in2

∫ 2π

0

∫ 1

0

r2n−1dr ∧ dθ = −2iπn.

On the other hand∫

D

dz ∧ dz = −2i

∫

D

rdr ∧ dθ = −2πi.

To prove the general case use the fact that f is a (branched) covering, and therefore

it behaves locally as the function z 7→ zn studied above.

Remark. The lemma, in a non-formal language, says that the area of f(Ω)

counted with “multiplicity” is equal to the “true” area of f(Ω) multiplied by the

degree of f .

3.2.5. We now prove that an annulus on a Riemann surface is always confor-

mally equivalent to a standard annulus on the complex plane.

Theorem (The Annulus Theorem). Let U be an open subset of R2 containing

the closed annulus z ∈ C; 1 ≤ |z| ≤ 2. Suppose that there exists a Riemann


surface structure on U such that the holomorphic functions (in that structure) are

smooth functions of R2. Then the open annulus

A = z ∈ C; 1 < |z| < 2,

with the complex structure induced from U , is biholomorphic to a unique annulus

AR = z ∈ C; 1 < |z| < R

with the standard Riemann surface structure induced from C.

Proof. It is easy to see (use 3.1.22) that there exists a barrier at every point

of ∂A so the Dirichlet problem has solution on A. For c a positive real number

let hc be the unique solution of the Dirichlet problem with boundary values 0 in

z; |z| = 1 and c in z; |z| = 2. Observe that hc is linear on c: if c, d and λ are

positive numbers, then hc+d = hc + hd and hλc = λhc. It is also easy to see that hc

is a proper function.

By Sard’s theorem (1.4.23) the set of points where the function h does have

zero derivative has measure zero. Let t be a regular value (the image of a point

where h has non-zero derivative); then h−1(t) is a collection of 1 dimensional closed

manifolds. By 1.4.24 these manifolds must be curves diffeomorphic to circles. By the

h−1(t)

Figure 12. h−1(t).

maximum modulus theorem for harmonic functions we have that none of this circles

can enclosed a disc in A. Otherwise the maximum and minimum of h in that disc

will be achieved in the boundary, where h is constant (with value t), and thus h will

be constant in the whole disc; but then, h will be constant on A by 3.1.7. Similarly,

we cannot have two (disjoint) curves in h−1(t) that bound an annulus inside A. So


we see that h−1(t) consists of a single curve, diffeomorphic to S1, and homotopic to

the boundary curves of A, as in figure 12. We simplify notation and write Ct for

this curve. We also have that if t < s then Ct and Cs bound a cylinder in A, with

Ct closer to S1 than Cs (see the remark after the proof for a formal definition of

“closer” in this setting).

Consider the integral λc =∫Ct∗dhc where 0 < t < c. From d ∗ dhc = ∆hc we see

that d ∗ hc is a closed form and therefore λc is independent of t (1.4.20). Since hc

depends linearly on c so does λc. In particular we have that either λc = 0 for all

c > 0, or λc → +∞ as c → +∞. If we write dhc = (∂hc/∂x)dx + (∂hc/∂y)dy,

we have (1.4.7) dhc ∧ ∗dhc = ((∂hc/∂x)2 + (∂hc/∂y)

2) dx ∧ dy. By the monotone

converge theorem we see that

0 <

∫

A

dhc ∧ ∗dhc = limǫ→0

∫

ǫ≤hc(z)≤c−ǫdhc ∧ dhc,

where we choose ǫ a regular value and such that 2 − ǫ is is also a regular value. By

Stokes’ theorem this last integral is equal to

limǫ→0

∫ Cc−ǫ

Cǫ

d(hc(∗dhc)) = (c− ǫ)

∫

Cǫ

∗dhc − (ǫ)

∫

Cǫ

∗dhc = cλc.

This implies that λc 6= 0. So there exists a unique value of c such that λc = 2π. Set

R = ec, and define a holomorphic function on A by the expression

f(z) = exp

(hc(z0) +

∫ z

z0

(dhc + i(∗dhc))),

where z0 is an arbitrary (but fixed) point of A. By our choice of c the periods of

the 1-form dhc + i(∗dhc) are integer multiples of 2πi, so f is well-defined. Observe

that |f | → 1 as |z| → 1, and |f | → R when |z| → 2. It follows that f : A → AR

is onto and proper. The surjectivity of f is a consequence of the fact that f is an

open mapping (it is holomorphic). To see that f is proper let K ⊂ AR be a compact

set. We can assume that K = z ∈ AR; r1 ≤ |z| ≤ r2, for 1 < r1 < r2 < R,

since any compact subset of AR is contained in one such annulus. Let ǫ > 0 be such

that 1 + ǫ < r1 and r2 < R − ǫ. Then we have that there exists a δ > 0, such that

|f(z)| ≤ 1 + ǫ, for z with |z| < 1+ δ, and |f(z)| ≥ R− ǫ, for z satisfying |z| > 2− δ.

Hence f−1(K) is contained in the annulus z ∈ A; 1 + δ ≤ |z| ≤ 2 − δ. Since

f−1(K) is closed it follows that it is also compact, and therefore f is proper, as we


claimed.

From 1.3.11 and exercise 21 we have that f is a (possibly branched) covering map,

of degree d ≥ 1. We need to show that d is precisely equal to 1. To see this we

use 3.2.4; first of all, a simple computation gives∫

A

df ∧ df = limǫ→0

∫

ǫ≤h(z)≤c−ǫd(fdf) = lim

ǫ→0−i

∫

ǫ≤hc(z)≤c−ǫd(|f |2(∗dhc)) =

= − 2πi(R2 − 1) =

∫

AR

dω ∧ dω.

Here ω = dx ∧ dy is the standard area form in the plane. The above computation

simply shows that the area of f(AR), counted with “multiplicity”, is the same as

the area of AR. Thus f has to be one-to-one.

Remark. The formal way of saying that Ct is closer to S1 than Cs is by saying

that Ct lies in the annulus bounded by Cs and S1.

3.2.6. Definition. A Riemann surface X is called planar is every smooth

closed 1-form on X with compact support is exact.

It is clear that any simply connected Riemann surface is planar: if ω is a form on

X, and p0 is a fixed point of X, the expression f(p) =∫ pp0ω defines a function of

X such that df = ω. Here the integration is done on a path from p0 to p; since X

is simply connected we have that the value of this integral does not depend of the

path. It is also clear that any open subset of a planar Riemann surface is planar.

Theorem. Let X be a Riemann surface, K a compact subset of X. Then there

exists a connected open subset U of X, with K ⊂ U , and a compact Riemann surface

Y such that U is biholomorphic to an open subset of Y . Moreover, if X is planar

then Y can be chosen to be planar.

Proof. Without loss of generality we can assume that K is connected. Choose

a smooth function with compact support, ϕ : X → R, such that ϕ(p) > 0 for all

p ∈ K. Let V = ϕ−1((0,+∞)), and r = infϕ(p); p ∈ K. Observe that r > 0

because K is compact. We have that ϕ : V → R+ is proper. Let E be the set of

critical points of ϕ in V. By Sard’s theorem (1.4.23) ϕ(E) has zero measure in R+;

and since ϕ is proper on V , this set ϕ(E) is closed in R+. Therefore there exist two


positive numbers, 0 < r1 < r2 < r, such that ϕ([r1, r2])∩E = ∅. Let c be a point in

the interval (r1, r2), and U the connected component of ϕ−1((c,+∞)) that contains

K. We will show that U satisfies the conditions in the statement of the theorem.

First of all the boundary of U is a collection of components of ϕ−1(c). Since ϕ

is proper, and c is not a critical value of ϕ, it follows from 1.4.24 that ∂U is a finite

collection of curves, Cini=1, where each curve Ci is diffeomorphic to the unit circle

S1. For each i = 1, . . . , n choose one such diffeomorphism, φ : Ci → S1, and extend

it to a smooth function ψi : Vi → S1, where Vi is a neighbourhood of Ci. One can

easily check that the Jacobian of the mapping

gi = (ψi, ϕ) : Vi → S1 × R

is never zero. Therefore there exists a neighbourhood Ti of Ci, Ti ⊂ Vi, and a

positive number ǫi, such that gi : Ti → S1× (c− ǫi, c+ ǫi) is a diffeomorphism. Fix δ

in (0, c); by the Annulus theorem (3.2.5) we have that there exists a biholomorphic

mapping hi : gi(S1 × (c − δ, c + δ)) → ARi

, where ARi= z ∈ C; 1 < |z| < Ri.

We can further assume that |hi| → Ri near g−1i (S1 × c − δ) and |hi| → 1 near

g−1i (S1 × c + δ). If that were not the case we only need to compose hi with the

mapping z 7→ Ri/z, which interchanges the two components of the annulus ARi.

We can thus use hi to attach (smoothly) the disc Di = z ∈ C; |z| < Ri to U ,

obtaining in this way a compact surface Y , that clearly contains a biholomorphic

copy of the set K.

To complete the proof of the theorem we need to show that Y can be chosen to be

planar whenX is a planar surface. Let ω be a closed 1-form with compact support on

Y . Since Di is simply connected, closed forms are exact, and therefore there exists a

smooth function fi on Di, such that w|Di= dfi. Let ai be a positive number; if ai is

small enough we can choose a smooth function χi, with compact support on Di, such

that χi ≡ 1 on the disc of radius Ri−ai (in Di), and ω′ = ω−∑i d(χifi) is a closed

1-form with compact support on X. Therefore we have that there exists a smooth

function g on X, with ω′ = dg on U . Consider the form ω′′ = ω′ − d((1−∑i χi)g).

We have now that ω′′ is a closed form with compact support in the disjoint union⊔iDi; hence ω′′ =

∑i ω

′′i , where ω′′

i is a closed form with compact support on Di.


Using again the fact that Di is simply connected we get functions Fi such that

ω′′i = dFi on Di, where Fi are smooth and have compact support on Di. Therefore

the form ω is exact, as we wanted to show.

3.2.7. Theorem. Any planar connected Riemann surface is biholomorphic to

an open subset of C.

Proof. If X is compact then we have that all forms have compact support.

The planarity condition implies that the space of holomorphic 1-forms have zero

dimension; that is, the surface has genus 0. We have already seen (2.3.5) that X

must be biholomorphically equivalent to the Riemann sphere C.

Suppose now that X is not compact. Since X is metrizable [3] we can write

it as an increasing union of connected open subsets Un, with compact closure. By

theorem 3.2.6 we have that each Un is biholomorphic to an open subset of a planar

compact Riemann surface Yn, which by the above remarks should be biholomorphic

to C. So we have a set of holomorphic, one-to-one (not necessarily surjective)

mappings fn : Un → C. Choose a point p in U1, and a holomorphic chart (U, z)

around p, with z(p) = 0 and U ⊂ U1. By replacing fn by anfn+bn, where an, bn ∈ C,

an 6= 0, we can assume that, for all n:

(1) fn(p) = 0; (2) and fn(p) = dz(p).

Let Kn be the set of one-to-one holomorphic functions on Un satisfying (1) and

(2). We have Kn are non-empty sets, and by Koebe’s theorem (3.2.1) each Kn is

compact. Hence the product K =∏

nKn is compact.

The sets Em = (g1, g2, . . .) ∈ K; gm|Un= gn, for n < m are non-empty and

closed. Since Em+1 ⊂ Em, we have that the intersection of all the Em’s is non-

empty. In other words, there exist holomorphic functions gm, defined on Um, such

that gm+1|Um= gm, on Um. So we have a holomorphic function g on X such that

g|Um= fm, for all m. Clearly g is on-to-one, and therefore it defines a biholomorphic

mapping from X onto an open subset of C.

3.2.8. Theorem (Poincare-Koebe Uniformization Theorem). Any simply con-

nected Riemann surface is biholomorphically equivalent to one (and only one) of the

following three surfaces: the Riemann sphere, the complex plane or the unit disc

(with their standard structures).


Proof. If X is simply connected then it satisfies the planarity condition, and

therefore X is biholomorphically equivalent to C or a simply connected open subset

of the complex plane. Using Riemann’s Mapping theorem, we have that, in the

latter case, X is biholomorphic to either C or the unit disc D.

It is not difficult to show that the Riemann surfaces C, C and D are not biholo-

morphic: first of all, the Riemann sphere is compact but the complex plane and the

unit disc are not. The complex plane and the unit disc are not biholomorphic since

any holomorphic function f : C → D must be constant, by Liouville’s theorem.

3.3. Uniformization of Riemann surfaces and

Kleinian groups

In this section we show that any Riemann surface can be written as a quotient

X/G, where X is a simply connected surface (studied in the previous section). The

elements of the group G are Mobius transformations; we study some properties of

these groups, which are elements of a big class of groups known as Kleinian groups.

3.3.1. Let X be a Riemann surface and π : X → X a universal covering.

From Topology (1.1.21) we have that X is homeomorphic to the quotient X/G,

where G is the group of deck transformations of the covering. The elements of

G are homeomorphisms of X, and non-identity transformations do not have fixed

points in X. In exercise 7 we asked the reader to show that X is a manifold. But

we have more than that: there is a (unique) Riemann surface structure on X such

that π becomes a holomorphic mapping. We have left the proof of this fact to

the reader, but we include it here because of its importance. Let p be a point of

X and U an evenly covered neighbourhood of p. By shrinking U if necessary we

can assume that there is a local coordinate defined on it, say z : U → z(U) ⊂ C.

Write π−1(U) =⊔j Vj as a disjoint union of open sets, where π|Vj

: Vj → U is a

homeomorphism. The mapping wj = z π|Vj: Vj → z(U) is a homeomorphism.

We take on X the atlas consisting of all local coordinates of this form, (Vj, wj).To show that X is a Riemann surface we only need to check that the changes of

coordinates are holomorphic mappings. But this is clear since changes of coordinates

3.3. UNIFORMIZATION OF RIEMANN SURFACES AND KLEINIAN GROUPS 149

on X are equal to changes of coordinates on X. More precisely, if (U , z) is another

local coordinate on X, with U evenly covered, let (V , w) be a local coordinate on X

constructed as above. Assume V ∩ Vj 6= ∅; then we have

wj w−1 = z (π|Vj) (π|eV )−1 z−1 = z z−1,

which is holomorphic since it is a change of coordinates on the Riemann surface

X. Observe that we have taken restrictions of the mapping π to sets where it is a

homeomorphism, so we can consider the inverse function (π in general will not have

a global inverse).

The expression of π in the above coordinates (U, z) and (Vj , w1) is given by

z π w−11 = z (π|Vj

) (π|Vj)−1 z−1 = Id : z(U) → z(U),

which shows that π is a holomorphic mapping.

The elements of G are homeomorphisms; moreover, they are biholomorphic map-

pings in the above Riemann surfaces structure. To prove this statement consider p

a point of X and g ∈ G. Let p1 = g(p) and denote by q the point q = π(p) = π(p1).

Choose a local coordinate (U, z) defined in a neighbourhood of q, with U evenly

covered, and let V0 and V1 the components of π−1(U) to which p and p1 be-

long, respectively. We have then local coordinates around these points given by

(V0, w0 = z (π|V0)) and (V1, w1 = z (π|V1)). Observe that g(V0) = V1 and

π|V1 g = π|V0. To see then that g is holomorphic we need to compose it with these

local coordinates:

w1 g w−10 = z (π|V1) g (π|V0)

−1 z−1 = Idz(U),

which proves our claim.

By the Uniformization theorem for simply connected surfaces (3.2.8) we have

that there exists a biholomorphic mapping f : Y → X, where Y is the Riemann

sphere, the complex plane or the unit disc. The mapping π : X → X is a covering if

and only if f π : Y → X is a covering, so we can assume that X is one of the three

mentioned surfaces. Putting all these facts together we obtain the general form of

the Uniformization theorem.


Theorem (Uniformization theorem for Riemann surfaces). Let X be a Rie-

mann surface. Then X is biholomorphic to X/G, where X is the Riemann sphere,

the complex plane or the unit disc, and G is a group of biholomorphisms of X,

isomorphic to the fundamental group of X.

Proof. We only need to show that X is biholomorphic to X/G. From Topology

we have that there exists a homeomorphism between these two surfaces; the proof

that such mapping is actually holomorphic is similar to the above computations so

we leave it to the reader.

3.3.2. Let p0 ∈ X and choose x0 ∈ X satisfying π(x0) = p0. Let U be an

evenly covered neighbourhood of x0 and π−1 =⊔j Vj as above. We have that

π−1(p0) = g(x0); g ∈ G, and if g and h are distinct elements of G then g(x0) and

h(x0) belong to different Vj ’s. Since these sets are disjoint we have that π−1(p0) is

a discrete subset of X (it does not have accumulation points). More precisely, if

there is a sequence of transformations, say gnn with gn(x0) → x1, then π(x1) =

limn π(gn(x0)) = limn π(x0) = p0. The point x1 belongs to one of the sets Vj, say

Vj1. But then we will have that gn(x0) ∈ Vj1 for n ≥ n0, a contradiction with the

fact that π restricted to Vj1 is a homeomorphism.

3.3.3. The group of automorphisms (biholomorphic self-mappings) of the Rie-

mann sphere, Aut(C), is the group of Mobius transformations, as we have seen in

corollary 1.3.14. We can identified Aut(C) with a group of matrices (or rather,

equivalence classes of matrices) as follows. Let GL(2,C) denote the group of square

matrices of order 2 with complex coefficients and non-zero determinant (equivalently,

the group of invertible linear mappings of C2). We define an equivalence relation ∼in this group by identifying M1 and M2 if there is a non-zero complex number, say

λ, such that M2 = λM1. The quotient space PGL(2,C) = GL(2,C)/ ∼ is known

as the projective general linear group. If we consider the subgroup SL(2,C) of

GL(2,C) of matrices of determinant equal to 1, and restrict the relation ∼ to com-

plex numbers λ with |λ| = 1, we obtain a quotient group PSL(2,C) = SL(2,C)/ ∼,

known as the special projective linear group. Since any non-zero number has a


square root in C it is not difficult to see that PGL(2,C) and PSL(2,C) are isomor-

phic groups. The identification between Aut(C) and PSL(2,C) is given by

Aut(C) → PSL(2,C)

A(z) =az + b

cz + d7→

a b

c d

,

where we use square brackets to denote equivalence classes of matrices in PSL(2,C).

It is easy to see that this mapping is a group homomorphism. From now onwards

we will freely interchange Mobius transformations with (classes of) matrices; for

example we will write the composition of two transformations as AB instead of the

more complicated notation A B.

3.3.4. Consider the map j : PSL(2,C) → P3 defined by

[a b

c d

]j7→ [a : b : c : d].

We can use j to put a topology on the group Aut(C): a sequence of Mobius trans-

formation Ann converges to the transformation A if and only if j(An) converges to

j(A). It is easy to see that this is equivalent to require that there exist elements of

PSL(2,C),

[an bn

cn dn

]and

[a b

c d

], corresponding to An and A respectively, such that

an → a, bn → b and so on.

Although this is the most natural topology of Aut(C) it does not behave nicely

with respect to the “character” of the transformations. For example, the sequence

of mappings An(z) = (1 + 1n) i zn converges to A(z) = i z; the transformation A is

a rotation around the origin, it preserves the circles centred at that point, but the

mappings An do not preserve any circle in the complex plane. Another example is

provided by the sequence of transformations Ann given by

An =

[(n+ 1)/n 1

0 n/(n+ 1)

].

Each of these mappings has two fixed points in C, zn = −n(n+1)2n+1

and ∞. The limit

of this sequence is A(z) = z + 1, which has only one fixed point, namely ∞ (the

sequence of fixed points zn converges to the point ∞, so in the limit all fixed points

“collapse”).


3.3.5. The number of fixed points can be used to classify Mobius transforma-

tions. We start with an easy lemma.

Lemma. A non-identity Mobius transformation has at least one and at most two

fixed points in C.

Proof. The fixed points of the transformation A(z) = az+bcz+d

are given by the

solutions of the equation A(z) = z in the Riemann sphere. If c 6= 0 we have a second

degree equation, az + b = cz2 + d, which can have at most two distinct roots (and

it has at least one). On the other hand, if c = 0 we can write A as A(z) = λz + µ,

where λ 6= 0. If λ = 1 the transformation A fixes only the point ∞; in the case of

λ 6= 1 the points ∞ and µ/(1 − λ) are fixed by A.

Corollary. If a Mobius transformation has three fixed points then it must be

the identity.

Assume A has only one fixed point, say z0. If z0 = ∞, then A is of the form

A(z) = z + µ. Let S be the transformation S(z) = 1µz (since we are assuming that

A has only one fixed point we have µ 6= 0); then SAS−1 is given by z 7→ z + 1. If

z0 6= ∞, the transformation S1(z) = −1z−z0 satisfies S1AS

−11 (z) = z + 1.

If A has two fixed points, say z0 and z1, let S2(z) = z−z0z−z1 , where we substitute

a factor (numerator or denominator) by 1 if the corresponding fixed point is the

point ∞. It is easy to see that (S2AS−12 )(z) = λz for some complex number λ, with

λ 6= 0, 1.

We can now give a classification of Mobius transformations.

Definition. Let A be a non-identity Mobius transformation. Then A is called

1. parabolic, if it is conjugate to z 7→ z + 1;

2. elliptic, if it is conjugate to z 7→ λz, where |λ| = 1 but λ 6= 1;

3. loxodromic, if it is conjugate to z 7→ λz, where λ 6= 0, 1. If λ is real and

positive A is called hyperbolic.

The above classification can be given in terms of the trace of the transformation as

the following lemma shows. The proof is an easy exercise left to the reader.

Lemma. Let A(z) = az+bcz+d

be a Mobius transformation with ad− bc = 1. Assume


A is not the identity transformation. Then:

1. A is parabolic if and only if (a + d)2 = 2;

2. A is elliptic if and only if (a+ d)2 < 4;

3. A is loxodromic if and only if (a + d)2 does not belong to the interval [0, 4].

In particular A is hyperbolic if and only if (a+ d)2 > 4.

Observe that A has order 2 if and only if a+ d = 0.

3.3.6. We can now look with more detail to some particular cases of the Uni-

formization theorem. We start with the easiest situation, when the universal cover-

ing is the sphere.

Proposition. If X is a Riemann surface whose universal covering space is

(biholomorphic to) the Riemann sphere then X is (biholomorphic to) the Riemann

sphere.

Proof. Non-identity covering transformations do not have fixed points, but

any Mobius transformation has at least two fixed points, so the covering group of

C → X must be trivial.

3.3.7. The next case we consider is that of surfaces covered by the plane. For

a biholomorphic mapping A : C → C we have that the point ∞ is a removable

singularity when we consider A as a mapping defined on the Riemann sphere (take

local coordinates and write A as a mapping from the punctured unit disc to itself).

If we extend A to C we have that A is a Mobius transformation fixing the point ∞so it must be of the form A(z) = λz + µ, with λ 6= 0 (see also [4, theorem 11.4, pg.

33]). In other words,

Aut(C) = g(z) = λz + µ; λ, µ ∈ C, λ 6= 0.

Assume that X is a Riemann surface covered by C and let G be the group of

covering transformations. Since the elements of G, other than the identity mapping,

cannot have fixed points, all transformations ofGmust then be of the form z 7→ z+µ.

If G is the trivial group then clearly X is the complex plane. Assume now that G

is cyclic; that is, it is of the form G = An(z) = z+nµ; n ∈ Z. If we conjugate G by

an automorphism of C, say S, we obtain that C/G and C/SGS−1 are biholomorphic


surfaces. Thus we can assume µ = 1. It is easy to see then that X is the punctured

plane, C∗ = z ∈ C; z 6= 0, and the covering mapping π : C → C∗ is given by the

exponential mapping, π(z) = e2πiz.

Suppose now that G has two generators; by a conjugation we can assume that

A(z) = z + 1 is an element of G. Let B(z) = z + µ be another element of G, not

in the subgroup generated by A. If µ = p/q is rational we can assume that p and q

are positive integers, with 0 < p < q and relatively prime. Let r and s be integers

such that r p + s q = 1. Then (AsBr)(z) = z + (1/q) and G will be cyclic. If µ is

real but not rational we can write µ = m+ ǫ, for some integer n and a positive ǫ in

(0, 1). Since the pair A(z) = z + 1, (A−mB)(z) = z + ǫ also generates G we can

assume that m = 0. For each positive integer n, there exists an integer pn, and a

non-rational number ǫn in (0, 1), such that nǫ = pn + ǫn. Consider the elements Cn

of G given by Cn = A−pnBn; these transformations are of the form Cn(z) = z + ǫn.

We claim that the numbers ǫn are all distinct: if ǫn = ǫm we will have ǫn = nǫ− pn

and thus mǫ = pm + nǫ − pn, which would imply that ǫ ∈ Q. Since all the ǫn are

distinct we can get a subsequence, say ǫnjj, converging to some point of [0, 1]. In

such case the transformations C−1nj+1

Cnjare all distinct and converge to the identity.

But then (C−1nj+1

Cnj)(z) → z for all z ∈ C, a contradiction with the definition of

covering space (see also 3.3.11). Hence we have that B(z) = z + µ, with µ not real;

we can assume that Im(µ) > 0 (take B−1 if necessary), and obtain that G is of the

form Gτ , as in example 1.3.6, so X is a torus.

We claim that these three cases, the complex plane, the punctured plane and

tori, are all the possibilities of Riemann surfaces covered by C. To prove the claim,

let X be a Riemann surface of the form C/G. All the transformations of G are of

the form Tλ : z 7→ z + λ. Let r = min|λ|; Tλ ∈ G. Observe that we take r to be

a minimum, not the infimum: if Tr were not in G we could construct a sequence of

distinct elements of G converging to the identity, using a trick similar to the above

one (we leave the proof to the reader). By a conjugation we can assume that r = 1.

Let µ be such that Tµ ∈ G and |µ| is minimum among the transformations in G

which are not of the form Tn(z) = z+ n, for n integer. If the group G is cyclic then

X should be either the complex plane or the punctured plane. On the other hand,


if G is not cyclic the argument above applies and we see that µ cannot be a real

number. Thus G contains a subgroup of the form Gµ. We claim that G = Gµ. If

not, let Tλ be an element in G but not in Gµ. Since 1, µ are linearly independent

over R we can write λ = r + sµ, where r and s are real numbers, but not integers.

Let m1 and m2 be two integers such that |r −m1| ≤ 1/2 and |s −m2| ≤ 1/2; the

number λ′ = λ−m1 −m2µ satisfies

|λ′| < 1

2+

1

2|µ| ≤ |µ|,

where the first inequality is strict since µ is not a real number. But this contradicts

the choice of µ. These computations complete the proof of the following theorem.

Theorem. If X is a Riemann surface whose universal covering space is C, then

X is (biholomorphic to) C, C∗ or a torus.

It follows from this theorem and 3.3.6 that “most” surfaces are covered by the

unit disc. In particular any compact surface of genus greater than 1 has D as its

universal covering. We will see some applications of this fact in the next section (for

example, the Riemann-Hurwitz theorem 3.4.20).

3.3.8. The next two results, which are easy consequences of Schwarz lemma,

characterise the automorphisms of the unit disc.

Lemma. If f : D → D is a biholomorphism of the unit disc with f(0) = 0. Then

f is a rotation around the origin; that is, f(z) = λz, for some complex number λ of

absolute value 1.

Proof. This result is part of Schwarz lemma (1.1.7).

Proposition. The automorphisms of the unit disc D are the Mobius transfor-

mations of the form Tw,λ(z) = λ z−w1−wz , where w ∈ D and |λ| = 1.

Proof. We first need to show that these transformations are automorphisms of

D. Let eiθ be a point of ∂D (the boundary of the unit dist); then we have

|Tw,λ(eiθ) = |λ|∣∣∣∣

eiθ − w

1 − we−iθ

∣∣∣∣ = |e−iθ|∣∣∣∣

eiθ − w

e−iθ − w

∣∣∣∣ = 1,

since the denominator of the last fraction is the complex conjugate of its numerator.

This computation only shows that Tw,λ(S1) ⊂ S1. Since T−1

w,λ(z) = T−w,λ we have


that Tw,λ(S1) = S1, and therefore T (D) is equal to either D of C\D. But since the

image of the origin is given by Tw,λ(0) = −λw, which is a point in D, we have that

Twλ(D) = D.

Let f : D → D be an arbitrary automorphism of the unit disc. Write w0 = f(0).

Then Tw0,1 f fixes the origin, so it must be a rotation by the previous lemma; that

is (Tw0,1 f)(z) = λz (|λ| = 1). A simple computation shows that

f(z) = T−1w0,1

(λ z) = Tλw,λ(z).

3.3.9. From the Riemann Mapping theorem (or the Uniformization theorem)

we have that the upper half plane H is biholomorphic to the unit disc D; the Mobius

transformation T (z) = z−iz+i

: H → D gives one such identification. To see this observe

that for x real we have that |T (x)| is the ratio of the distance from x to i to the

distance from x to −i and therefore |T (x)| = 1. By topological arguments we get

that T (R ∪ ∞) = S1 and T (H) must be either the unit disc or its exterior. Since

T (i) = 0 we have that T (H) = D. The advantage of using the upper half plane

over the unit disc is that many computations are easier. For example, the next

proposition shows that the automorphisms of H are just Mobius transformations

with real coefficients and positive determinant, certainly simpler expressions than

those of elements of Aut(D).

Proposition. The automorphisms of H are the Mobius transformations of

the form A(z) = az+bcz+d

, where a, b, c, d are real numbers satisfying ad − bc > 0 (or

equivalently ad − bc = 1). The group Aut(H) acts transitively on H; that is, for

any two points w0 and w1 of H, there exists an element T ∈ Aut(H), such that

T (w0) = w1.

Proof. If A : H → H is an automorphism of the upper half plane, then TAT−1

is an automorphism of the unit disc, where T (z) = z−iz+i

. By 3.3.8 we have that TAT−1

is a Mobius transformation and therefore A is also a Mobius transformation. This

shows that Aut(H) is a group of Mobius transformations.

Let G denote the group of Mobius transformations of the form given in the state-

ment of the proposition; we want to show that G is the full group of automorphisms


of H. If A ∈ G we have that

A(z) =az + b

cz + d=az + b

cz + d

az + b

cz + d=ac|z|2 + adz + bcz + bd

|cz + d|2 ;

so

Im(A(z)) =(ad− bc) Im(z)

|cz + d|2 .

This shows that G is a subgroup of Aut(H) (if z has positive imaginary part so does

A(z)).

Let w0 = x0+iy0 be a point of the upper half plane. The transformationM(z) = z−x0

y0

satisfies M(w0) = i (since w0 ∈ H we have y0 > 0). We can write M as

M(z) =

z√y0

− x0√y0√

y0

,

so M belongs to G (in the above expression we have taken the positive square root of

y0, which is possible since y0 is a positive real number). Therefore G acts transitively

on H (map w0 to i and then i to w1), and consequently Aut(H) too.

If B is an element of Aut(H) fixing the point i the transformation R = TBT−1 is

an automorphism of D that fixes the origin. Hence R(z) = λ2 z, for λ a complex

number of absolute value 1. The matrices corresponding to R and T are

R =

λ 0

0 λ

and T =

1√2i

1 −i

1 i

,

respectively. If we write λ = cos(θ) + i sin(θ) an easy calculation shows that B is

given by

B(z) =cos(θ) z + sin(θ)

− sin(θ) z + cos(θ),

which belongs to G.

Let now C denote any automorphism of H; we have C(w0) = i for some point

w0 in the upper half plane. Let M be as above. Then MC−1 fixes the point i, so it

follows from the above computation that MC−1 ∈ G. Since M ∈ G we have that

Aut(H) = G.

In a way similar to the identification of Aut(C) with PSL(2,C) we can give an iso-

morphism between Aut(H) and PSL(2,R), where this last group consists of equiv-

alence classes of matrices with real coefficients and determinant 1. In this case


we do not have that PSL(2,R) and PGL(2,R) are isomorphic, since a matrix in

GL(2,R) with negative determinant cannot be equivalent to a matrix with positive

determinant (negative numbers do not have square roots in R).

3.3.10. What elements of Aut(H) have fixed points in H? First of all, if A ∈PSL(2,R) fixes the point z0 ∈ C, then A must also fixed its conjugate z0, since the

coefficients of A are real (we understand ∞ = ∞). If A is parabolic its fixed point

must be in R = R ∪ ∞. If A is elliptic, with ad − bc = 1, the solutions of the

equation A(z) = z are given by

z =a− d±

√(d− a)2 + 4bc

2c=a− d±

√d2 + a2 − 2ad+ 4bc

2c=

=a− d±

√(a+ d)2 − 4

2c.

Since c 6= 0 and 0 ≤ (a+ d) < 4 the transformation A must have a fixed point in H.

If A is loxodromic it must be hyperbolic and it is easy to see that its fixed points

are both in R. It follows from this computations that if X is a Riemann surface of

the form X = H/G then G does not have elliptic elements.

3.3.11. We next define a general class of groups of Mobius transformations

and reformulate the Uniformization theorem.

Definition. A group of Mobius transformation G is said to act properly

discontinuously at a point z ∈ C if there exists an open neighbourhood U of z,

such that the subgroup of G given by

g ∈ G; g(U) ∩ U 6= ∅

is finite. We denote by Ω(G) the (open) set of points of the Riemann sphere where

G acts properly discontinuously; this set is called the region of discontinuity of

G. The group G is called Kleinian if Ω(G) 6= ∅.

We can now rewrite the Uniformization theorem in terms of Kleinian groups.

Theorem (Uniformization theorem). Any (connected) Riemann surface X is

biholomorphic to a quotient of the form X/G, where X is the Riemann sphere, the

complex plane or the unit disc, and G is a Kleinian group satisfying X ⊂ Ω(G).

Remark. Observe that in this above result we do not claim that X is equal


to the region of discontinuity Ω(G) of G. There are cases when these two sets

are different. For example, if X is the punctured unit disc D∗, then X = H and

G = z 7→ z+n; n ∈ Z. But the region of discontinuity of G is the whole complex

plane and C/G is the punctured plane C∗ (the upper half plane covers the punctured

unit, the lower half plane the exterior of the unit disc and R covers S1).

3.3.12. Proposition. Kleinian groups are discrete.

Proof. By discrete we mean that G does not have accumulation points in

PSL(2,C), with the topology described in 3.3.4. Assume that G is not discrete;

then there exists a sequence of distinct elements of G, say Ann, such that An →A, where A is a Mobius transformation, not necessarily in G. The sequence of

Mobius transformations Bn = A−1n+1Ann has infinitely many distinct elements and

converges to the identity. But then Bn(z) → z, for all z in C, and therefore G

cannot be Kleinian.

LetG be a Kleinian group, A an elliptic transformation ofG. By conjugating with an

element of PSL(2,C) if necessary we can assume that A is of the form A(z) = eiθ z.

It is easy to see that A has finite order if and only if θ is a rational number. Assume

now that θ /∈ Q, and define a mapping j :< A >→ S1, where < A >= An; n ∈ Zis the subgroup of G generated by A, by the expression j(An) = ein θ. Since A does

not have finite order we get that the image of j is an infinite set of S1 and therefore

it has an accumulation point, say eiθ0 . Let njj be a sequence of integers such that

einj θ → eiθ0 . Then the transformations Anjconverge to z 7→ ei θ0 z, so G cannot be

Kleinian. We have proved the following result.

Proposition. If G is a Kleinian group and A is an elliptic element of G then

A has finite order.

3.3.13. Another interesting property of Kleinian groups is given in the follow-

ing proposition.

Proposition. A Kleinian group is either finite or infinite countable.

Proof. Let z0 be a point in the region of discontinuity of G and H the stabiliser

of z0 in G; that is,

H = g ∈ G; g(z0) = z0.


Since G acts properly discontinuously at z0 we have that H is finite. Let G(z0)

denote the orbit of z0 under the given group, G(z0) = g(z0); g ∈ G. We have that

G(z0) is a discrete set of C and it must then countable (exercise 83). On the other

hand, it is easy to see that there is a bijection between G/H and G(z0), given by

[g] 7→ g(z0). It follows that G is either finite or infinite countable.

3.3.14. Our next application of the Uniformization theorem is to determine

all surfaces with abelian fundamental group.

Lemma. Let A and B be two Mobius transformations, neither of them equal to

the identity. Assume that AB = BA. Then one and only one of the following cases

is satisfied:

1. if A is parabolic then B is also parabolic and they have the same fixed points;

2. if A is not parabolic then B is not parabolic and either they have the same

fixed points, or both transformations have order 2, and each of them interchanges

the fixed points of the other.

Proof. First of all, the results of the lemma are invariant under conjugation,

so we can choose the fixed points of the transformations in a way that computations

are easy. Observe that if A fixes a set W pointwise (that is, A(w) = w for all w in

W ), then B fixes W as a set, B(W ) = W , although B does not need to fix each

point of W . Clearly this statement holds if we interchange A and B.

Assume first that A is parabolic, say A(z) = z + 1 (remember that we are free

to conjugate A and B for our computations). Then B(∞) = B(A(∞)) = A(B(∞))

so B(∞) must be a fixed point of A, which implies that B(∞) = ∞, and hence B

is of the form B(z) = λz + µ. If B fixes a point z0 in C, from the above remark we

see that A must fix the point z0, which cannot happen by hypothesis. Hence λ = 1

and B is a parabolic transformation with fixed point ∞.

Assume now that neither of the transformation is parabolic (by the above com-

putation, if one transformation is parabolic so is the other). Let A be of the form

A(z) = λz. From the above computations we have that B(0,∞) = 0,∞, so

there are two possible cases:

1. B(∞) = ∞ and B(0) = 0. Then A and B have the same fixed points.

2. B(∞) = 0 and B(0) = ∞. In this case B(z) = µ/z. By a conjugation


that does not change A we can assume that µ = 1. Then B fixes ±1. Since

A(1,−1) = 1,−1 we see that A(z) = −z (the possibility of A being the identity

does not occur by hypothesis), so A has order 2 and this completes the proof.

3.3.15. Theorem. Suppose X is a Riemann surface with abelian fundamental

group. The one (and only one) of the following cases occurs:

1. X is simply connected and X is C, C or D;

2. π1(X, x0) ∼= Z and X is C∗, D∗ or Ar = z ∈ C; r < |z| < q, for some real

number r ∈ (0, 1);

3. π1(X, x0) ∼= Z ⊕ Z and X is a torus.

Remark. In the above theorem all statements have to be understood “up to

biholomorphisms”.

Proof. The first case is the Uniformization theorem for simply connected sur-

faces; the surfaces with universal covering space the complex plane have been studied

in 3.3.7. Thus we have only to study surfaces with abelian fundamental group and

the upper half plane as the universal covering space. Moreover, we can assume that

the fundamental group is not trivial.

If π1(X, x0) is cyclic, then X = H/ <A>, where A is an element of PSL(2,R).

Since non-trivial deck transformations do not have fixed points A will be either

parabolic or hyperbolic. In the first case we can assume that A(z) = z + 1, after

a conjugation and taking inverses if necessary. Then X = D∗ and the covering

mapping is z 7→ exp(2πiz). In the case of A hyperbolic we have A(z) = λz, for some

number λ > 1; we get that X is an annulus, with covering mapping

z 7→ exp

(2πi

log z

log λ

),

where log is the principal branch of the logarithm, log(reiθ) = log r+ iθ. The radius

of the annulus is given by r = exp(−2π2

log λ) ∈ (0, 1).

To complete the proof of the theorem we need to show that there are no Riemann

surfaces with abelian fundamental group of rank greater than 1 and universal cover-

ing space the upper half plane. If one of the generators of G, say A, is parabolic, we

can assume that A(z) = z+1. By lemma 3.3.14 all elements of G are also parabolic,

and because they are automorphisms of H, they must be of the form z 7→ z + t,


for t real. But when we studied surfaces covered by C we saw that in that case G

would not be discrete. If A is hyperbolic, we have that all elements of G are also

hyperbolic; a similar proof shows that this case cannot occur.

3.4. Hyperbolic Geometry, Fuchsian Groups and

Hurwitz’s Theorem

In this section we will study some properties of groups of automorphisms of

the upper half plane. We show that there exists a natural metric on H, called

the hyperbolic metric, for which the elements of Aut(H) are isometries. It follows

from this that we can put a metric on compact surfaces (of genus greater than 1).

A somehow surprising result is that the area of a surface does not depend of the

Riemann surface structure. We will also prove that the group of automorphisms of

compact surfaces (covered by H) is finite.

3.4.1. If γ : [a, b] → R2 is a piecewise smooth curve (1.4.18) its length in

Euclidean geometry is given by the integral

||γ||E =

∫ b

z

|γ′(t)| dt.

(Since γ is piecewise smooth the integral is finite.) This statement is usually for-

mulated by saying that the infinitesimal length element (the length of the tangent

vector γ′) is |dz|. The distance dE between two points p0 = (x0, y0) and p1 = (x1, y1)

is the length of the segment joining them, that is:

d(p0, p1) = ||s||,

where

s(t) = (tx0 + (1 − t)x1, ty0 + (1 − t)y1), for 0 ≤ t ≤ 1.

One can check that the length of this curve is minimum among the lengths of the

curves joining p1 and p2; that is,

dE(p0, p1) = ||s|| = inf||γ||; γ(a) = p1, γ(b) = p2.

3.4. HYPERBOLIC GEOMETRY, FUCHSIAN GROUPS AND HURWITZ’S THEOREM 163

The segments are called geodesics. It can be easily verified that the distance between

any two points in a geodesic is given by the length of the piece of the geodesic joining

them.

3.4.2. By the above argument we see that to define a metric on the upper half

plane it suffices to give its infinitesimal length element. We set this to be equal to

ds = |dz|Im(z)

. As explained in the case of the Euclidean metric, this simply means

that the length of a (piecewise smooth) curve γ : [a, b] → H is given by the following

expression:

||γ|| =

∫ b

a

|γ′(t)|Im(γ(t))

dt.

This integral is finite because the curve is assumed to be piecewise smooth. Similarly

one defines the distance d between two points z0 and z1 of H by

d(z0, z1) = inf||γ||; γ(a) = z0, γ(b) = z1.

We will use d for this new distance and dE for the (standard) Euclidean distance. ds

and d are called the hyperbolic metric and hyperbolic distance, respectively.

We need to prove that d is indeed a distance, but before that we will study some

properties of the metric ds and its relation with Mobius transformations.

3.4.3. Before proceeding further we recall some results from Complex Analysis.

Definition. Let zj, j = 1, . . . , 4 be four distinct points in C; the cross ratio

of these points is defined by

(z1, z2; z3, z4) =z4 − z2z4 − z1

z3 − z1z3 − z2

,

where we delete the corresponding terms (or we take limits) if one of the points is

the point ∞.

Observe that in the case of one of the four points being ∞ there will be two terms

in the above expression with ∞ in them, one in the numerator and the other in the

denominator, so after removing those terms we are left with a well defined fraction.

Some authors change the order of the factors in the definition of cross ratio. However,

for the applications all definitions are equivalent. It can be easily proved that of the

possible 24 definitions of cross ratio (there are 24 permutations of four letters) there

are only 6 different values.


If z1 = ∞, z2 = 0 and z3 = 1 then (z1, z2; z3, z4) = z4. More generally, an easy

computation show that S(z) = (z1, z2; z3, z) is the value (at z) of the unique Mobius

transformation S that takes z1, z2 and z3 to ∞, 0 and 1, respectively. This remark

will be useful in the proof of the following result.

3.4.4. Lemma. Mobius transformations preserve cross ratios. More precisely,

if T is a Mobius transformation, and zj, j = 1, . . . , 4 four distinct points in the

Riemann sphere, then (T (z1), T (z2);T (z3), T (z4)) = (z1, z2; z3, z4).

Proof. The proof can be done with an easy (but long) direct calculation;

however, with the last remark in the above subsection we can get a short and elegant

proof as follows. Let S be the Mobius transformation that takes z1, z2 and z3 to ∞,

0 and 1, respectively (this S is given by S(z4) = (z1, z2; z3, z4)). Then ST−1 takes

T (z1), T (z2) and T (z3) to ∞, 0 and 1 respectively and therefore we have

(T (z1), T (z2);T (z3), T (z4)) = ST−1(T (z4)) = S(z4) = (z1, z2; z3, z4).

3.4.5. Lemma. For distinct points in the Riemann sphere lie on a line or

circle if and only if their cross ratio is real.

Proof. We will show first that the image of R = R ∪ ∞ under a Mobius

transformation is a line or a circle. If S(z) = (z1, z2; z3, z) is given by S(z) = az+bcz+d

,

then S(z) is real if and only if S(z) = S(z); that is,

az + b

cz + d=az + b

cz + d.

From this expression we obtain

(6) (ac− ac)|z|2 + (ad− bc)z + (bc− ad)z + (bd − bd)z = 0.

If ac− ac = 0 then we must have ad− bc 6= 0. Otherwise we get the following pair

of equations

ac = ac, ad = bc.

If a 6= 0 we have d = bca, and thus ad−bc = a bc

a= a bc

a−bc = 0, which is not possible.

On the other hand, if a = 0 we get bc = 0, which again gives us ad − bc = 0. So


we see that ad − bc 6= 0. Write ad − bc = u + iv, z = x + i and bd = r + is; then

equation (6) becomes

vx+ uy + s = 0.

Since u and v cannot be simultaneously equal to 0 we get that this is the equation

of a line.

If ac− ac 6= 0 equation (6) is equivalent to

∣∣∣∣z +ad− bc

ac− ac

∣∣∣∣ =

∣∣∣∣ad− bc

ac− ac

∣∣∣∣ ,

which the the equation of a circle.

To complete the proof of the lemma we argue as follows. If (z1, z2; z3, z4) is a real

number then zj lies in S−1(R), where S is the Mobius transformation that defines

the cross ratio (i.e. S(z) = (z1, z2; z3, z)). By the first part of the proof we have that

S−1(R) is either a line or a circle.

Suppose now that (z1, z2; z3, z4) lie in a line or circle, say C. If we consider the

transformation S once more we have that S−1(0), S−1(1) and S−1(∞) are in C, so

S−1(R) = C, and therefore S(C) = R. Thus S(z4) = (z1, z2; z3, z4) is a real number

(it cannot be ∞ by the definition of cross ratio).

Corollary. If C is the family of lines and circles in C and A is a Mobius

transformation, then A(C) = C.

3.4.6. The Mobius transformation T (z) = z−iz+i

: H → D can be used to define

a hyperbolic metric on the unit disc such that T becomes an isometry. This means

that if the metric on D is given by λ(z) |dz|, where λ is a positive function, then we

must have

λ(T (z)) |T ′(z)| =1

Im(z),

for z in H. If that is the case a simple use of the change of variables theorem for

integrals shows that ||γ|| = ||T (γ)|| for a piecewise smooth curve on H. It is easy to

see that λ is given by the expression

λ(z) =2

1 − |z|2 .


This allows us to switch between the upper half plane and the unit disc when we

prove results regarding the hyperbolic metric.

3.4.7. Proposition. Aut(H) acts by isometries with respect to the hyperbolic

metric: for any piecewise smooth curve γ : [a, b] → H and any Mobius transforma-

tion A ∈ Aut(H), one has ||γ|| = ||A(γ)||.Proof. Let A be given by A(z) = az+b

cz+d, where the coefficients are real and

satisfy ad − bc = 1. We have A′(z) = 1(cz+d)2

. In 3.3.8 we computed that

Im (A(z)) =Im(z)

|cz + d|2 .

Using these expressions we get

|A′(z)|Im(A(z))

=1

Im(z),

so

||A(γ)|| =

∫ b

a

|A′(t)| |γ′(t)|Im(A(γ(t))

dt =

∫ b

a

|γ′(t)|Im(γ(t))

dt = ||γ||.

3.4.8. Theorem. d is a distance in H. The topology induced by it is the

standard Euclidean topology.

Proof. It is easy to see that d is symmetric, non-negative and satisfies the

triangle inequality. We will show that d(w0, w1) is strictly positive for w0 6= w1. We

will work in the unit disc, since the computation is easier in this case, and in the

process we will obtain a formula for the distance of a point in D to the origin that

will be useful later.

Let w0 and w1 be two distinct points in D. Using the Mobius transformation

M(z) =z − w0

1 − w0z

we can assume that w0 = 0. By a rotation we can further assume that w1 = t, where

t is a point in the open interval (0, 1). Consider a path γ : [0, 1] → D joining 0 and

t. If we write γ(t) = x(t) + iy(t), we have that

||γ|| =

∫ 1

0

2|γ′(t)|1 − |γ(t)|2 dt ≥

∫ 1

0

2|x′(t)|1 − x(t)2

dt ≥∫ 2x(t)

1−x(t)2

0

dt = log

(1 + t

1 − t

).


In the second inequality we have used that the function f(s) = 11−s2 is negative for

s < 0 and increasing for s ≥ 0. Observe that the hyperbolic length of the path

γ(s) = st, s ∈ [0, 1], is precisely the above expression, log(

1+t1−t); thus d(0, t) =

d(w0, w1) > 0. This completes the proof of the fact that d is a distance.

To prove that the topology induced by d is the standard topology of D we use

the above computations. We have that the hyperbolic disc Dh(0, r), of centre 0 and

radius r > 0, is given by

Dh(0, r) = z ∈ D; |z| < er − 1

er + 1;

that is, an Euclidean disc of centre 0 and different radius. This shows that the

neighbourhoods of 0 in the hyperbolic and Euclidean topologies are the same. Since

the group of Mobius transformations acts transitively by homeomorphisms in D

(3.3.8) we have that both topologies are the same.

As a corollary of the above computations we get that the distance from 0 to any

point w ∈ D is given by

d(0, w) = log

(1 + |w|1 − |w|

).

In particular, as w approaches S1 (in the Euclidean distance) we have that d(0, w)

goes to infinity. This shows that the unit circle is at infinity distance of any point

in the unit disc (apply the triangle inequality). Similarly the real axis is at infinite

(hyperbolic) distance from points in the upper half plane. Thus the hyperbolic

metric is the natural one if we want to study properties of H (or D) on its own,

rather than considering it as a subset of the Riemann sphere.

3.4.9. A geodesic is a (smooth) curve that minimises the distance locally

between points in it (its image). More precisely, if γ is a geodesic defined on the

interval (a, b) and t0 ∈ (a, b), then there is a neighbourhood U of γ(t0), such that the

distance between any two points in γ((a, b))∩U is given by the length of γ between

those two points. For example, if γ is a geodesic in the upper half plane, with the

same notation we have that if γ(tj) ∈ U , for j = 1, 2,

d(γ(t1), γ(t1)) =

∫ t2

t1

|γ′(t)|Im(γ(t))

dt.


However a geodesic does not need to minimise distances globally. Consider the case

of the sphere S2 where geodesics are given by great circles, that is, the intersection

of planes through the origin with S2. If p1 and p2 are two points in the sphere, not

diametrically opposed, then there are two geodesics joining them, one of which will

realize the distance between p1 and p2 while the other hand will have longer length.

We also have that there could be more than one geodesic between two points. In

the same example of the sphere, any two points diametrically opposed are joined

by infinitely many different geodesics. And there are spaces where some points can-

not be joined by geodesics. The space R2\(0, 0) with the Euclidean metric is an

example; the points (1, 0) and (−1, 0) are at distance 2, but there is no geodesic

between them realizing that distance. In the case of the hyperbolic metric we are in

the best possible situation: any two points can be joined by a unique geodesic that

realizes the distance between them.

3.4.10. Proposition. The hyperbolic geodesics of D are the circles and lines

perpendicular to S1. In the case of the upper half plane, the geodesics are the circles

and lines perpendicular to the real axis. Given any two points in D (or H) there

exists a unique geodesic between them; moreover, such geodesic realizes the distance

between any two points in its image.

Proof. In the proof of 3.4.8 we have obtained that the segment (0, 1) is a

geodesic in D. It is not difficult to see that the full diameter (−1, 1) is a geodesic.

If C is a circle (or line) perpendicular to S1 it is possible to find an automorphism

of the unit disc, say A, such that A(C ∩ D) = (−1, 1) (exercise 77). It follows from

this that any circle or line orthogonal to S1 is a geodesic.

Conversely, if γ is a geodesic in the unit disc, consider two points wj = γ(tj), j = 0, 1,

close enough so that γ realizes the distance between them. By an automorphism of

D, say A, we can map w0 to 0 and w1 to a point in (0, 1). Since the automorphisms

of D are hyperbolic isometries we have that γ should be contained in the image of

(−1, 1) under A−1 and thus it is a circle or line orthogonal to S1 (to be more precise,

the image of (a, b) under γ is contained on a circle or line orthogonal to S1).


To show that any two points of D lie in a unique geodesic it suffices to consider

that case of one point being the origin and the other a point t in the interval (0, 1).

But again this is part of the proof of 3.4.8.

3.4.11. Theorem. The hyperbolic metric on D (H) is complete.

Proof. Let znn be a Cauchy sequence in the unit disc with respect to the

hyperbolic metric. Given ǫ > 0 there exists an n0 such that d(zn, zm) < ǫ, for

n,m ≥ n0. Therefore d(0, zn) ≤ d(0, zn0) + d(zn0 , zn), and thus d(0, zn) is bounded.

The distance d(0, zn) is given by log(1+|zn|1−|zn|), so there exists a number 0 < R < 1,

such that |zn| ≤ R, for all n. Since the set z; |z| ≤ R is compact in the Euclidean

topology there exists a convergent subsequence. But the topologies induced by the

hyperbolic and Euclidean metrics are equivalent, so that subsequence converges (to

the same limit point) with respect to the hyperbolic metric. The triangle inequality

shows that the full sequence znn converges in the hyperbolic metric.

3.4.12. The angle between two lines or circles in H meeting at a point z0 is

defined as the angle formed by the tangent lines to the curves at z0. For simplicity

we say that two lines or circles meeting at a point of R do it with angle equal to 0. A

triangle is the portion of H enclosed by three distinct geodesic that meet pairwise.

A triangle is called ideal if the geodesics meet in a point in the (extended) real axis.

The hyperbolic area of a region D of H is given by the integral

Area(D) =

∫

D

1

y2dx dy.

Theorem (Gauss-Bonet for triangles). The hyperbolic area of a triangle with

angles α, β and γ is equal to π − (α+ β + γ).

Proof. Consider first the case of a triangle T with two angles equal to 0.

By using a Mobius transformation and the reflection r(z) = −z (which is also a

hyperbolic isometry and preserve angles, see exercise 78) we can assume that T is

as in figure 13. In this case we can compute the area directly as follows:

Area(T ) =

∫∫

T

1

y2dx dy =

∫ d

0

∫ ∞

√c2−(x−c)2

1

y2dx dy =

=

∫ d

0

1√c2 − (x− c)2

dx =

∫ α

0

−dθ = π − α.


If T has only one angle equal to 0 we can compute its area as the difference of the

area of two triangle, each of them with two zero angles, as in the figure 14. The

general case follows in a similar way.

0 c d

∞

α

T

Figure 13. Triangle with two zero angles.

β

α

β

γ

δ

α

TS

Figure 14. Gauss-Bonet.

3.4.13. We want now to apply some of these results to compact surfaces cov-

ered by the unit disc. We start with a definition.

Definition. A Kleinian group G is called Fuchsian if there exists a disc or

half plane H which is invariant under the elements of G, that is, g(H) = H for all

g ∈ G.

When talking about Fuchsian groups we will use the word disc to mean a disc or a

half plane.

A striking fact of Fuchsian groups is that discreteness and properly discontinuous

action are almost equivalent.

Proposition. Let G be a Fuchsian group with invariant disc H. The following

are equivalent:


1. G acts discontinuously on H (i.e. H ⊂ Ω(G));

2. G is discrete.

Proof. We have seen that Kleinian groups are discrete, so we only need to

prove “2 ⇒ 1”; we will show this by contradiction. By a conjugation we can assume

that D is the unit disc and G does not act properly discontinuously at the origin.

This means that for any neighbourhood U of 0, the set g ∈ G; g(U) ∩ U 6= ∅ is

infinite (in particularG is not finite, which is obvious since finite groups acts properly

discontinuously on the whole Riemann sphere). Let r1 be a positive number and

define U = D(0, r1), in the hyperbolic metric. Let z1 and w1 be points in U\0,such that z1 = g1(w1), for some transformation g1 ∈ G. Choose a positive number

r2 satisfying r2 < mind(z1, 0), d(w1, 0). We can find z2 and w2 in U\0, such

that z2 = g2(w2), for g2 ∈ G, and g2 6= g1. Continuing this process we find sequences

of positive numbers rnn, points in the unit disc zn and distinct elements of G

gnn, such that:

(1). rn is a decreasing sequence converging to 0;

(2). d(0, zn) < rn;

(3). d(g−1n (zn), 0) < rn.

Since G acts by isometries on the hyperbolic distance we have

d(0, gn(0)) ≤ d(0, zn) + d(zn, gn(0)) = d(0, zn) + d(g−1n (zn), 0) < 2rn.

Let wn = gn(0). By proposition 3.3.8 we have that gn is of the form

gn(z) = λn

(z + wn1 + wnz

),

where |λ| = 1. Choose a subsequence λnjj with λnj

→j λ0. Since wn → 0 we have

that the transformations gnjconverge to the rotation R(z) = λ0z, and thus G is not

discrete.

3.4.14. The action of a Kleinian group on its region of discontinuity (or a part

of it) is better understood by taking a set that contains one element of each orbit.

For example, if G is the group of translations G = Tn(z) = z + n; n ∈ Z, acting

on H, we have that every point of H can be mapped by an element of G to a point z,

with 0 ≤ Re(z) < 1. If we consider the vertical strip S = z ∈ H; 0 ≤ Re(z) ≤ 1,


we have that G identifies the two vertical lines in the boundary of this strip. The

quotient space H/G is equivalent to S/G; it is easy to see (geometrically) that S/G

is a cylinder, which is clearly homeomorphic to the punctured disc. We have indeed

proved that H/G is the punctured disc; this discussion might help us to understand

why. The next definition generalises this situation to Fuchsian (or Kleinian) groups.

Definition. Let G be a Fuchsian group acting on H. A fundamental domain

of G for its action on H is a connected open set D satisfying the following conditions:

FD1: for every element g of G, not equal to the identity, g(D) ∩D = ∅;FD2: for every z in H there exists a transformation g of G, such that g(z) belongs

to D, the closure of D in H;

FD3: the boundary of D in H consists of a countable number of smooth curves,

called the sides ofD. For every side s there exists another side, say s′, not necessarily

distinct from s, and an element g of G, such that g(s) = s′ and (s′)′ = s;

FD4: (local finiteness) for every compact set K of H, the group

g ∈ G; g(K) ∩K 6= ∅,

is finite.

3.4.15. The following result is needed to prove the existence of fundamental

domains.

Lemma. Let G be a non-cycle Fuchsian group with invariant disc ∆. Then there

exists a point z0 ∈ ∆ that is not fixed by any non-trivial element of G.

Proof. Assume ∆ = D and that 0 is fixed by some non-trivial element of G;

let H be the subgroup of G consisting of the elements that fix 0. By lemma 3.3.8

all elements of H are rotations around the origin. If G = H by discreteness we

have that G must be cyclic. On the other hand, if H is a proper subgroup, since

G acts properly discontinuously at 0, we can find a positive number r, such that

g(U)∩U = ∅, for all g /∈ H , where U is the disc of centre 0 and radius r. Any point

in U\0 will satisfy the conditions of the lemma.

3.4.16. Let G be a Fuchsian group that leaves the upper half plane invariant

and choose p ∈ H satisfying the conditions of the above lemma. For g ∈ G, not


equal to the identity, define

Hg = z ∈ H; d(p, z) < d(g(p), z),

where we use the hyperbolic metric to measure distances. Thus Hg consists of the

points in H that are closer to p than to g(p). Geometrically one can obtain Hg by

considering the hyperbolic geodesic segment that joins p and g(p), say L, and then

taking the geodesic L′ orthogonal to L on its midpoint; Hg will be the half plane

determined by L′ containing p. The Dirichlet region Dp(G) of G (relative to p) is

defined as the intersection of all such hyperplanes:

Dp(G) =⋂

g 6=IdHg.

For example, if G is the group of translations G = Tn(z) = z + n; n ∈ Z, and

p = (1/2) + i, the Dirichlet region Dp(G) is precisely the (open) vertical strip we

considered in 3.4.14: Dp(G) = z ∈ H; 0 < Re(z) < 1.Proposition. The Dirichlet region is a fundamental domain for the action of

G on H.

Proof. To simplify notation we will write D for the Dirichlet region Dp(G)

(relative to some point p).

Condition FD1. Let g 6= Id be an element of G and z a point of D. Since

z ∈ Hg−1 we have

d(g(p), g(z)) = d(p, z) < d(g−1(p), z) = d(p, g(z)),

which implies that g(z) does not belong to D.

Condition FD2. Let z be a point in the upper half plane. By the discontinuous

action of G we have that there exists an element g ∈ G (not necessarily unique),

such that d(g(p), z) ≤ d(h(p), z) for all h ∈ G. If we write the elements of G as g hwe have

d(g−1(z), p) = d(z, g(p)) ≤ d(z, (g h)(p)) = d(g−1(z), h(p)).

This means that g−1(z) belongs to the closure of Hh for all h ∈ G and therefore to

the closure of D.

Condition FD3. Since G is countable it is clear that the boundary of D has at


most countably many sides. Let z be a point in the relative interior of a side s. This

is equivalent to say that there exists a unique element g ∈ G, such that

d(z, p) < d(z, h(p)), ∀h 6= g, Id and d(z, p) = d(z, g(p)).

Hence d(g−1(z), p) = d(g−1(z), g−1(p)) = d(z, p), and for h 6= g−1, Id we have

d(g−1(z), h(p)) = d(z, (g h)(p)) > d(z, p) = d(g−1(z), p).

Thus g−1(z) belongs to the side s′ with g−1(s′) = s.

Condition FD4. Let K ⊂ H be compact. Without loss of generality we can

assume that K is the closed disc centred at p and of hyperbolic radius r (any compact

subset of the upper half plane is contained in one such disc). We have that there are

only finitely many images of p (under transformations of G) in the disc of radius 2r

centred at p. From this it follows that if d(g−1(p), p) > 2r then g(D)∩K = ∅.

3.4.17. The following lemma is easy to prove; it shows that the Dirichlet region

is a “good” choice of fundamental domain.

Lemma. Let G be a Fuchsian group acting on H and D a Dirichlet region of G.

1. The quotient surface H/G is compact if and only if D is compact in H.

2. If D is compact then D has only finitely many sides.

3.4.18. Assume that X is a Riemann surface given by H/G; that is, X is

biholomorphic to the quotient surface H/G. We can put a metric on X by using

the natural projection π : H → X similar to the way we calculated the metric on D

from the mapping T : H → D. Although π does not need to be globally one-to-one,

it is so locally, and that is all we need. More precisely, for a point p0 in X, let

U be an evenly covered neighbourhood of p0, and write π−1(U) =⊔j Vj for the

decomposition of the preimage of U in disjoint open sets of H, each homeomorphic

via π to U . The functions zj(π(p)) = p, for p ∈ Vj , serve as local coordinates on X.

Thus to define a metric on X all we need to do is to find expressions of the form

λj(π(p)) |dzj|, such that λj = λk

∣∣∣dzk

dzj

∣∣∣. We then set λj(π(p)) = 1Im(p)

, for p ∈ Vj. For

any other set Vk as above, there exists an element g ∈ G, such that g(Vj) = Vk. The

value of λk is given by λk(π(q)) = 1Im(q)

, and zk(π(q)) = q = g(p) = g(zj(p)), for


q ∈ Vk, so p = g−1(q) belongs to Vj. Under these circumstances we have that

1

Im(q)=

1

Im(g(p))=

|g′(p)|Im(p)

=1

Im(p)

∣∣∣∣dzkdzj

∣∣∣∣

We call this metric the hyperbolic metric of the surface. Observe that the metric

depends on the complex structure of X; however, the area of X does not, as the

following result shows.

Theorem (Gauss-Bonet for compact surfaces). If X is a compact surface of

genus g ≥ 2 then the hyperbolic area of X is equal to 2π (2g − 2).

Proof. Let D be a Dirichlet region for G. By 3.4.17 we know that D has only

finitely many sides, so its boundary has zero area. On the other hand, by property

FD1 we have that π : D → X is one-to-one, so Area(D) = Area(X). Choose a point

p0 in the interior of D and join it to the (finitely many) vertices of the boundary

of D by geodesics. This is possible, since D is a finite intersection of convex sets,

and thus it is convex. In this way we obtain a triangulation of D that projects to a

triangulation of X. Assume that the sides and vertices of ∂D project to E sides and

V vertices on X. It is not difficult to see, using the Euler-Poincare formula, that

V − E + 1 = 2 − 2g. On the other hand, by the Gauss-Bonet theorem for triangles

we have

Area(D) = 2Eπ − 2π −∑

(interior angles at the vertices) .

The term −2π comes from the sums of the angles at p0. Since the vertices of ∂D

project to V points in X, we have that the sum of the interior angles is 2πV , and

therefore

Area(X) = Area(D) = 2π(2g − 1 + V ) − 2π = 2π (2g − 2).

3.4.19. Our next goal is to study automorphisms of compact surfaces. Let

π : H → X be the universal covering of a compact surface of genus g ≥ 2, with

covering group G. If f : X → X is an automorphism we can lift it to a biholo-

morphic mapping A : H → H; in particular A is a Mobius transformation. The


transformation A satisfies πA = fπ; for any element g ∈ G we have

π A g A−1 = f π g A−1 = f π A−1 = f f−1 π = π,

so there exists an element h ∈ G, such that AgA−1 = h. In other words, A belongs

to N(G), the normaliser of G in Aut(H). (The group N(G) is the biggest subgroup

of Aut(H) on which G is normal; it consists of the element B ∈ Aut(H), such that

BGB−1 = G). The converse statement is also true; namely, if B ∈ N(G), then

the expression h(π(z)) = π(B(z)) defines an automorphism, h, of X. Since the

elements of G will project to the identity mapping of X, we can identified Aut(X)

with N(G)/G. The following result guarantees that (under mild conditions on G)

the group N(G) is Fuchsian.

Proposition. If G is a torsion-free non-cyclic Fuchsian subgroup of PSL(2,R),

then N(G) is also Fuchsian.

Recall that a group is said to be torsion-free if there are no non-trivial elements of

finite order.

Proof. All we need to show is that N(G) is discrete (3.4.13). Assume that

there exists a sequence hn of distinct elements of N(G) converging to the identity.

For all g ∈ G we have that hngh−1n n is a sequence of elements of G converging to

g. Since G is Fuchsian we get that hngh−1n = g, for n > n0, for some positive integer

n0. By lemma 3.3.14 we get that hn and g must have the same fixed points (since

we are assuming that G is torsion-free, the situation where hn and g have order 2

does not occur).

If all elements of G have the same fixed points we would have that G consists of only

parabolic or hyperbolic transformations. In either case G would be cyclic, against

the hypothesis. Let us choose g1 and g2 in G with at least three distinct fixed points,

say z1, z2 and z3. Then every element h of N(G) will fix zj , for j = 1, 2, 3. By 3.3.5

we get h = Id.

3.4.20. The automorphisms group of the Riemann sphere is the group of

Mobius transformations. For the case of a torus Tτ , any translation of the form

T (z) = z + c, with c ∈ C, induces an automorphism on Tτ . Thus in these two cases


we have that the automorphisms group is not only infinite, but it is not discrete

either. In the case of compact surfaces this cannot happen.

Theorem (Hurwitz). Let X be a compact Riemann surface of genus g ≥ 2.

Then Aut(X) has at most 84(g − 1) elements.

Proof. If X = H/G, since G is torsion-free and non-cyclic we know that N(G)

is Fuchsian (3.4.19), and thus Y = H/N(G) is a Riemann surface. The covering

H → Y clearly factors through X, so Y must be compact. The mapping q : X → Y

has degree equal to the order of H = Aut(X), say n; thus H is a finite group

(remarks 1 and 2 below). To find the bound on n we make a detailed study of the

Riemann-Hurwitz formula.

The set of points of an automorphism of X (other than the identity) are finite,

and since H is finite as well, we have that the set of points of X fixed by some

non-trivial element of H is a finite set. Let p1, . . . , pr be a maximal set of inequiv-

alent fixed points of non-trivial elements of H . That is, each pj is fixed by some

automorphism of X not equal to the identity; and if j 6= k, we have that h(pj) 6= pk

for all h ∈ H . Thus these points project under q to different points of Y . For each

j, let νj be the order of the subgroup Hj of H of automorphisms of X fixing pj. We

have that there are n/νj distinct points in X that project to the same point, q(pj)

of Y , and each such point is fixed by a subgroup of H of order νj . Thus we obtain

that the total branching number B of the mapping q is given by

B =

r∑

j=1

n

νj(νj − 1) = n

r∑

j=1

(1 − 1

νj

).

Observe that νj ≥ 2, so 1(1/νj) ≥ 1/2. The Riemann-Hurwitz formula in this

setting gives us

2g − 2 = n

(2γ − 2 +

r∑

j=1

(1 − 1

νj

)),

where γ is the genus of Y (and g ≥ γ). If g = γ then n = 1 (recall that n is the

order of Aut(X), which we are trying to bound). In the case of g > γ we have the

following cases:

• γ ≥ 2. Then 2(g − 1) ≥ 2n implies that n ≤ g − 1.


• γ = 1. In this case we have a value νj ≥ 2, so the right hand side of the

Riemann-Hurwitz relation is, at least, equal to n/2, or equivalently, n ≤4(g − 1).

• γ = 0 and r ≥ 5. This cases gives n ≤ 4(g − 1).

• γ = 0 and r = 4. Since the right hand side of Riemann-Hurwitz relation must

be positive we get that at least one νj ≥ 3, and n ≤ 12(g − 1).

• γ = 0, r = 3. We can assume that 2 ≤ ν1 ≤ ν2 ≤ nu3. Then ν3 > 3 and

ν2 ≥ 3. There are several cases to study:

a. If ν3 ≥ 7 we get n ≤ 84(g − 1), with equality in the case of ν1 = 2, ν2 = 3

and ν3 = 7.

b. ν3 = 6, ν1 = 2. Then ν2 ≥ 4 and n ≤ 24(g − 1).

c. ν3 = 6, ν1 ≥ 3. Then n ≤ 12(g − 1).

d. ν3 = 5, ν1 = 2. Then ν2 ≥ 4 and n ≤ 40(g − 1).

e. ν3 = 5, ν1 ≥ 3. Then n ≤ 15(g − 1).

f. ν3 = 4, ν1 ≥ 3. Then n ≤ 24(g − 1).

Remarks. 1. If G is a Fuchsian group then H/G is a Riemann surface. 2. It

is easy to show that, in the situation of the above proof, there exists a point p ∈ X

which is not fixed by any non-identity element of Aut(X). This shows that the order

of the covering X → Y = X/Aut(X) is equal to the order of the group Aut(X).

3.5. Moduli spaces

3.5.1. So far in this book we have studied properties of a fixed Riemann surface.

The problem of moduli spaces deals with the study of varying Riemann surface

structures on a fixed topological surface. More precisely, two surfaces X and Y

are said to be conformally equivalent (or simply equivalent) if there exists a

biholomorphic mapping between them, f : X → Y . Our goal is to know under what

conditions X and Y are equivalent. An example of this type of problem is given by

the Uniformization theorem (3.2.8); it classifies all simply connected surfaces up to

biholomorphisms. The general problem is difficult and the study of it constitutes

3.5. MODULI SPACES 179

a whole area of research on its own, with new mathematical tools. In this section

we will give a couple of examples of how this problem can be treated; the reader

interested on more results can find a nice introductory text in [18].

3.5.2. Before we get to explain our examples we need to make a few general

remarks on the relation between conformally equivalent surfaces and their universal

coverings and covering groups. Let X and Y be two surfaces, with universal cov-

erings X and Y , and covering mappings πX and πY respectively. Let f : X → Y

be the lift of f to the universal covering spaces; the following diagram will be then

commutative:

Xf

//

πX

Y

πY

Xf

// y

In particular we have that f is a Mobius transformation (see §§ 1.3.14, 3.3.3

and 3.3.9). The spaces X and Y are homeomorphic; we will then identify them

and consider X as the universal covering space of both X and Y .

We can also give a more algebraic statement, in terms of the covering groups.

Let GX and GY be the covering groups of X and Y respectively. Since we have

identified the universal covering spaces of X and Y we can consider these two groups

as subgroups of Aut(X). Then the mapping f will satisfy fGX f−1; that is, GX and

GY are conjugate subgroups of Aut(X). We will use this formulation of the problem

in our examples since it make many of the computations easier.

3.5.3. Our first example consists on the study of equivalence classes of annuli.

Let r1, r2 and r be real numbers satisfying r1 < r2 and 1 < r; we denote by

A(z0, r1, r2) the annulus z ∈ C; r1 < |z−z0| < r2 and by Ar the annulus A(0, 1, r).

Clearly A(z0, r1, r2) is equivalent to Ar2/r1 by the transformation z 7→ 1r1

(z−z0). So

every annulus is conformally equivalent to one of the form Ar, which means that the

space of equivalence classes of annuli is contained in the interval (1,+∞). To fully

determine conformal equivalence of annuli we need to find under what conditions

Ar and As (with r and s real numbers greater than 1) are equivalent. The universal

covering of Ar is given by πr : H → Ar, where πr(z) = exp (2πi log z/ log λ), and r


and λ are related by the expression r = exp (−2π2/ log λ) (3.3.15). Here log is the

principal branch of the logarithm defined on C\[0,+∞). The covering group Gr is

generated by the transformation gr(z) = λ z; that is, Gr = z 7→ λn z; n ∈ Z.Let Gs denote the covering group of the annulus As, and let gs(z) = µ z be a

generator of Gs. If f : Ar → As is a biholomorphism, and f : H → H a lift to the

universal covering space, then f will be an element of SL(2,C). Since f conjugates

Gr into Gs we have that f grf−1 is equal to either gs or g−1

s . The transformation

M(z) = −1/z is an automorphism of H that conjugates gs into g−1s , so we can

assume that f grf−1 = gs (otherwise we consider Mf , which is also a lift of f). The

fixed points of gr and gs are 0 and ∞, so f(0,∞) = 0,∞. But since gnr (z0) → ∞for z0 in H and n → +∞ we must have f(∞) = ∞, and therefore f(0) = 0. So f

is of the form f = k z, for k a positive real number. A simple computation shows

that, with this expression of f , we have f grf−1 = gr, which mean that r = s. Thus

we have proved the following result.

Theorem. Any annulus A(z0, r1, r2) is conformally equivalent to one and only

one annulus of the form Ar, for r > 1. More precisely we can take r = r2/r1.

Remark. See [22, pg. 291] for a purely analytic proof of the above theorem.

3.5.4. Consider now the case when X and Y are surfaces of genus 1. By the

Abel-Jacobi theorem (2.9.13) they must be of the form C/Gτ = Tτ , for some τ with

positive imaginary part. The classification of tori is given by the next result.

Theorem. Two tori Tτ and Tη are conformally equivalent if and only if

(7) τ =aη + b

cη + d,

where a, b, c and d are integer numbers satisfying ad− bc = 1 (that is, τ and η are

related by an element of SL(2,Z)).

Proof. Let us denote by T τn,m the transformation z 7→ z+n+mτ (and similarly

for η). Suppose f is a mapping satisfying fGτ f−1 = Gη. Since f is an automorphism

of C it must be of the form f(z) = λ z+µ. It is easy to check the following identities:

f T τ1,0 f−1(z) =z + λ

f T τ0,1 f−1(z) =z + λτ.


We must then have λ = cη+d and λτ = aη+ b, for some integers a, b, c and d; that

is, τ and η satisfy the relation 7. Since f−1Gηf = Gτ , the transformation z 7→ az+bcz+d

must be invertible; that is, ad − bc = ±1. The imaginary part of aη+bcη+d

is equal to

ad−bcIm(η)

, so we must have ad− bc = 1. This proves one half of the theorem.

Assume now that τ and η are related by an element of SL(2,Z), as in the theorem.

Let S be the Mobius transformation given by S(z) = cz + d. If c = 0 then a = d =

±1, which implies that τ = η so there is nothing to prove. Thus we can assume that

c 6= 0. It is not difficult to see that

ST τn,mS−1 = T ηnd+mb,nc+ma ;

this equation implies that SGτS−1 ⊂ Gη. Choosing (n,m) = (a,−c) and (n,m) =

(−b, d) we get that SGτS−1 contains the transformations T η1,0 and T η0,1, and thus

SGτS−1 ⊃ Gη. So the transformation S conjugates Gτ into Gη and therefore the

tori Tτ and Tη are conformally equivalent.

3.5.5. From the above theorem we have that the space of equivalence classes of

tori, denoted by M1 can be identified with H/SL(2,Z). To study this space we can

follow the techniques of 3.4. It is not difficult, for example, to find a fundamental

domain for the action of SL(2,Z) on H. Let P be the open polygon bounded by the

geodesics:

L1 = z ∈ H; Re(z) = 1/2 ,

L2 = z ∈ H; Re(z) = −1/2 ,

L2 = z ∈ H; |z| = 1 .

Claim. P is the Dirichlet region (for the action of SL(2,Z) on H) centred at the

point 2i.

Proof of the claim. We first prove that L1 is contained in the set of points

equidistant from 2i and 2i + 1; that is, L1 ⊂ DA for A(z) = z + 1, in the nota-

tion of 3.4.16. Observe that since d(z, 2i) = d(z, 2i + 1) we have d(R(z), R(2i)) =

d(R(z), R(2i + 1)) for R(z) = 12

+ iz. But then we have

d(R(z), 2i + 1) = d(R(z), R(2i)) = d(R(z), R(2i + 1)) = d(R(z), 2i).


In other words, if d(z, 2i) = d(z, 2i + 1) then d(R(z), 2i) = d(R(z), 2i + 1). Since

R(z) = z if and only if Re(z) = 12

we have that L1 ⊂ DA (the geodesic DA is the

“full” vertical line containing L1).

Similarly one can prove that L2 ⊂ DA−1 and L3 ⊂ DB, where B(z) = −1/z. These

three statements show that D2i is contained in P .

Assume now that D2i is a proper subset of P . Then there exists a point z0 ∈ P

and a non-trivial element h ∈ SL(2,C), such that h(z0) ∈ P . Write h(z) = az+bcz+d

.

We have (see 3.3.9) Im(h(z0)) = Im(z0)|cz0+d|2 . Write z0 = x0 + iy0. Since z0 is in P we

have

|z0|2 = x20 + y2

0 > 1 and−1

2< x0 <

1

2.

Using these inequalities one can easily prove the following:

|cz0 + d|2 = c |z0|2 + 2 c d x0 + d2 > |c|2 + |d|2 − |cd| = (|c| − |d|)2 + |cd|.

The last term in the above displayed formula is a positive integer (it cannot be 0

since ad − bc 6= 0). Thus it is at least equal to 1, which implies that |cz0 + d|2 > 1.

So we have that if z0 and h(z0) are both in P then Im(h(z0)) < Im(z0). We can

apply he same argument to h(z0) and z0 = h−1(h(z0)), to get Im(z0) < Im(h(z0)).

This contradiction shows that P is indeed equal to D2i.

To have a picture of the space of (conformal equivalence classes) of tori we just need

to consider P , the closure of P in the hyperbolic plane, by the action of SL(2,Z)

(observe that the point of infinity is not a part of P ). The transformation A identifies

L2 and L1, while B fixes the side L3 (as a set, not necessarily pointwise). Thus we can

think of ∂P as consisting of four sides, L1, L2, s = z ∈ H; |z| = 1, 0 ≤ Re(z) < 12

and s′ = z ∈ H; |z| = 1, −12< Re(z) ≤ 0, with B(s) = s′ (condition FD3 in

definition 3.4.14). Any torus will be conformally equivalent to one torus of the form

Tτ where τ belongs to P . The precise formulation is given in the next result.

Theorem. Any torus is conformally equivalent to one and only one torus Tτ

with τ satisfying the following conditions:

1. |τ | ≥ 1;

2. −12< Re(τ) ≤ 1

2;

3. If |τ | = 1 then Re(τ) ≥ 0.


3.5.6. The boundary of P has three vertices: i, 1+i√

32

and −1+i√

32

(the point i

is the meeting point of the sides s and s′). The tori corresponding to these points

are special in the sense that we explain next.

An automorphism M : C → C of the complex plane induces an automorphism on

a torus Tτ if and only if MGτM−1 = Gη, where τ and η are related by an element

of SL(2,Z). We observe that if M is of the form M(z) = z + µ, for µ a complex

number, then MGτM−1 is actually equal to Gτ . Moreover, if µ is not of the form

n +mτ (for n and m integers), then M induces a non-trivial automorphism of Tτ .

Thus we have that any torus has a group of automorphism with “many” elements

(see the remarks before Hurwitz’s theorem in 3.4.20). The mapping M(z) = −zalso conjugates Gτ into itself, so it will give another automorphism of Tτ . The fixed

points of M are given by points z0 of C satisfying M(z0) = z0 + n + mτ , for some

integers n and m. It is easy to see that there are only four possible points, up to

equivalence by elements of Gτ : 0, 1/2, τ/2 and (1 + τ)/2. So if f : Tτ → Tτ denotes

the automorphism of the torus Tτ induced by M , we see that f has four fixed points

on Tτ , corresponding to the above four points. In particular one can prove that f is

the “hyperelliptic involution” of Tτ (we use quotation marks since we have defined

hyperelliptic involutions only for surfaces of genus at least 2).

Suppose τ ∈ P corresponds to a torus with some automorphism different from

the ones in the previous paragraph. Then there exists a Mobius transformation

S(z) = az+bcz+d

, in SL(2,Z), such S(τ) = τ . Solving this equation we get that τ should

be of the form

τ =a− d

2c+ i

√4 − (a + d)2

2c.

We have that |a + d| < 2 (otherwise τ would be real) and c 6= 0 (since z 7→ z + µ

does not have fixed points on P ). Since |τ | ≥ 1 we get 1−adc2

≥ 1. This inequality

gives us the following different options:

a = 0, c = 1, d = 0, τ1 = i;

a = 1, c = 1, d = 0, τ2 =1 + i

√3

2;

a = 1, c = −1, d = 0, τ3 =−1 + i

√3

2;


Consider the torus Tτ1 , and let T1 be the Mobius transformation T1(z) = τ1z(= iz).

We have T 41 = Id and T1T

in,mT

−11 = T i

m,−n. Thus T1 induces an automorphism of

order 4 on Tτ1 . The tori corresponding to τ2 and τ3 are conformally equivalent (by

the transformation z 7→ z − 2), so we consider only one of them, say Tτ2 . Using

the identity τ 32 = −1 we see that T2(z) = τ2z satisfies T2T

τ2n,mT

−12 = T τ2m.n−m. Since

the mapping (p, q) 7→ (q, p − q) of Z2 is invertible we have that T2 induces an

automorphism on Tτ2 . It is easy to check that the order of that automorphism is 6.

We obtain that the tori corresponding to the vertices of P are precisely those with

some “extra automorphisms”.

3.5.7. Topologically all annuli are “the same”, that is, homeomorphic, and

similarly all tori. If one takes two annuli and identifies the boundaries (“glue”

them by their boundaries) one gets a torus. This particular surface has an order 2

mapping interchanging the two annuli. Such mapping cannot be holomorphic (in

the torus) since it has many fixed points, namely the two curves that formed the

boundaries of the two annuli. However, it is possible to show that this mapping is

anti-holomorphic (that is, its conjugate is holomorphic). More precisely, let τ = it

be a complex number with t > 1, and consider the symmetry (anti-holomorphic

mapping of order 2) of the complex plane given by σ : z 7→ −z. We have

σT τn,mσ−1(z) = σT τn,m(−z) = σ(−z + n + it) = z − n− it = T τ−n,−m(z).

Thus σ induces an automorphism of Tτ , say R. It is easy to check that R is anti-

holomorphic and has order 2. What are the fixed points of R? If a point p in Tτ

is fixed by R, then there exist integers n and m such that z0 = −z0 + n + mti,

where z0 is a point of C that projects to p under the natural quotient map. The

solutions of this equation are given by the lines Ln = z ∈ C; Re(z) = n2. Since Ln

is equivalent under elements of Gτ to Ln±2n′ we have only two set of solutions, the

imaginary axis and L1. These two lines project to two closed curves on Tτ (since

z 7→ z + ti belongs to Gτ ), the ones corresponding to the boundaries of the annuli

above explained.

Consider now the transformation ρ(z) = −z; it is clear that ρ induces an anti-

holomorphic involution on H. If S is an element of SL(2,C), then ρ S ρ = S, so


ρ induces a mapping on M1, the space of tori. If τ is fixed point of ρ then there

exists an element T ∈ SL(2,C) such that ρ(τ) = T (τ). But then T ρ T−1 will be

an anti-holomorphic mapping of H fixing τ . So, without loss of generality we can

assume that ρ(τ) = τ . The solutions of this equation are given by τ ∈ H satisfying

Re(τ) = 0; that is, the tori “built by gluing two annuli”.

187

Exercises

Exercise 1. Let f : C → C be a holomorphic function. Assume that there exists

a constant M , such that Re(f(z)) ≤ M , for all z ∈ C. Show that f is constant.

Exercise 2. Prove Liouville’s theorem: a bounded holomorphic function defined

on the complex plane, f : C → C, is constant.

Exercise 3. Prove the Fundamental Theorem of Algebra: any polynomial with

complex coefficients has at least one root in C.

Exercise 4. Let f : D → C be a holomorphic function defined on an open set

of the complex plane. Write f = u + iv (u and v are the real and imaginary parts

of f , respectively). Using the Cauchy-Riemann equations show that u and v are

harmonic functions.

Exercise 5. Prove by direct computation that if X is a simply connected surface

then H1(X,Z) is trivial.

Exercise 6. Show that any connected, locally path connected topological space is

path connected.

Exercise 7. Let p : X → Y a covering. Show that X is a manifold if and only if Y

is a manifold.

Exercise 8. Let G be a group and H = [G,G] its commutator subgroup; that is,

H is generated by all elements of the form ghg−1h−1, for g and h in G. Prove that

H is a normal subgroup and G/H is abelian.

Exercise 9. Show that the punctured plane C∗ is homeomorphic to a cylinder.

Exercise 10. Prove the following version of Van Kampen Theorem:

let M be a topological space, U and V open subsets of M satisfying M = U ∪ V

and such that the intersection U ∩ V is path connected and non-empty. If U and V

are simply connected then M is simply connected.

Use this result to show that C is simply connected.

Exercise 11. Consider the following subspace of R2: X = A ∪B ∪ C, where:

188

A = (x, y); x ≥ 0, y = 1, B = (x, y); x ≥ 0, y = −1;C = (x, y); x < 0, y = 0.Define a topology on X by putting the subspace topology on A\(0, 1), B\(0, 1)and C, and for the points (0, 1) (and (0,−1)) use the basis of neighbourhoods given

by Nǫ = (x, 1); 0 ≤ x < ǫ ∪ (x, 1); −ǫ ≤ x < 0 (similarly for (0,−1)). Show

that X is locally homeomorphic to R but it is not a manifold.

Exercise 12. Let p : X → Y be a covering mapping, where X and Y are man-

ifolds, and let p∗ : π1(X, x0) → π1(Y, p(x0)) be the induced mapping between the

fundamental groups. Show that p∗ is injective.

Exercise 13. Find an example of a covering map p : X → Y , and a continuous

function f : X → X such that f p = f but f is not a homeomorphism (this shows

that the condition of homeomorphism cannot be removed from the definition on

1.1.20.

Exercise 14. Show that the result of exercise 7 is true if you substitute manifold

for Riemann surface.

Exercise 15. Can you find a Riemann surface structure on C which is not compat-

ible with the usual structure, induced by (C, Id)?

Hint : C is homeomorphic to D.

Exercise 16. Prove that the stereographic projection P : S2 → C, given by

P (x1, x2, x3) =

x1+ix2

1−x3, if x3 6= 1,

∞, if x3 = 1,

is a homeomorphism between the sphere S2 of R3 and the Riemann sphere C.

Exercise 17. The projective line (or projective space of dimension 1) P1 is defined as

follows: consider the equivalence relation ∼ in M = C2\(0, 0) given by (z1, z2) ∼(w1, w2) if there exists a non-zero complex number λ, such that z1 = λw1 and

z2 = λw2. Set P1 = M/ ∼. Prove that C is homeomorphic to P1.

Hint : denote by [z : w] the points P1 (notation as in 2.7.1), consider the map from

P1 to C given by [z : w] 7→ z/w if w 6= 0 and [z : 0] 7→ ∞.

Exercise 18. Let G be a group of Mobius transformations and D a connected

open subset of C. Assume that the following conditions are satisfied:

189

1. g(D) = D, for all g ∈ G.

2. For every point z in D, there exists a neighbourhood U of z, such that, if

g ∈ G satisfies g 6= ID, then g(U) ∩ U = ∅.Denote by D/G the quotient space of D by the action of G; that is, z1 and z2 are

equivalent if there exists g ∈ G, such that g(z2) = z1. Show that D/G carries a

unique Riemann surface structure such that the quotient mapping π : D → D/G is

holomorphic.

Exercise 19. Show that if f : X → C is holomorphic and X is a compact Riemann

surface then f is constant.

Exercise 20. Show that the set of poles of a holomorphic function defined on a

Riemann surface is discrete.

Exercise 21. A mapping f : X → Y between topological spaces is called proper

if f−1(K) is compact in X, for any K compact subset of Y . Extend the definition

of degree and proposition 1.3.11 to the case of proper mappings (not necessarily

between compact surfaces).

Exercise 22. Prove proposition 1.3.14

Exercise 23. Let f : X → Y be a holomorphic mapping between compact

Riemann surfaces of genera g and g′ respectively. Show that if g = g′ then f is a

covering map. Show that if g′ = 1 then g = 1.

Exercise 24. Prove that the residue of a polynomial p(z) = a0 + a1z+ · · ·anzn at

the point ∞, when one considers p as a meromorphic mapping on C, is equal to a1.

Exercise 25. Prove that a meromorphic function f : C → C on the Riemann

sphere is rational, i.e. the quotient of two polynomials.

Exercise 26. Let R(z) = p(z)q(z)

be a rational function on the Riemann sphere, where

p and q are polynomials without common factors. Show that the point ∞ is a:

• pole if deg (p) > deg (q);

• zero if deg (p) < deg (q);

• regular point if deg (p) > deg (q).

Exercise 27. Let f : X → C be a non-constant meromorphic function on a

Riemann surface. For a point p of X, let n denote the ramification index of f at

190

that point. Define

z(q) =

(f(q) − f(p))1/n, if f(p) 6= ∞,

(f(q))−1/n, if f(p) = ∞,

for q is a neighbourhood of p. Show that z is a local coordinate on X.

Exercise 28. Let X be a compact surface, p1, . . . , pn distinct points of X. Prove

that any non-constant holomorphic function f : X ′ = X\p1, . . . , pn → C comes

arbitrarily close to any point z ∈ C. More precisely, show that for any z ∈ C, and

any ǫ > 0, there exists a point p ∈ X ′ such that |f(p) − z| < ǫ.

Exercise 29. Let τ and µ be two complex numbers with positive imaginary parts.

Let Tτ and Tµ denote the corresponding tori (see 1.3.6). Show that if λ is a complex

number satisfying λGτ ⊂ Gµ, then the function f : C → C, given by f(z) = λz,

induces a holomorphic mapping from F : Tτ → Tµ. Prove that F is biholomorphic

if and only if λGτ = Gµ.

Exercise 30. Let f : X → Y be a non-constant holomorphic mapping between

Riemann surfaces. Show that the mapping f ∗ : O(Y ) → O(X), defined by f ∗(g) =

g f , is a ring homomorphism.

Exercise 31. Let Tτ = C/Gτ be as in 1.3.6. Show that Tτ is a connected Hausdorff

space. Moreover, show that Tτ is a topological manifold of dimension 2.

Exercise 32. Show that the operators ∂ and ∂ satisfy ∂ ∂ = 0 and ∂ ∂ = 0, by

direct computation.

Exercise 33. Let α be a smooth (complex valued) 2-form on a surface X with

compact support. Let U = Uini=1 and V = Vjmj=1 be two finite open coverings

of the support of α by coordinate patches. Let fnni=1 and gjmj=1 be partitions of

unity subordinated to U and V respectively. Show that

n∑

i=1

∫

X

fiα =

m∑

j=1

∫

X

gjα.

Exercise 34. Let X = X/G be a Riemann surface, where X is the universal

covering space and G the group of deck transformations. Let p : X → X be the

covering map. Assume ω is a closed 1-form on X, and F a primitive of p∗(ω), that

is, F : X → C satisfies dF = p∗(ω). Show that for every g in G there exists a

191

complex number ag, such that F − F g−1 = ag.

Exercise 35. Use the Residues theorem to show that there does not exist a

function on C/Gτ with a just pole of order 1.

Exercise 36. LetG be a group with at least two elements, andX a topological space

with at least two non-empty disjoint open sets. Define a presheaf F by F(U) = G

if U ⊂ X is a non-empty open set, and F(∅) the trivial group. The restriction

homomorphisms ρ : F(U) → F(V ) are given by the identity homomorphism if

V 6= ∅ and the trivial homomorphism if V = ∅. Show that F is a presheaf but not

a sheaf. Compute the associated sheaf.

Hint : consider two disjoint open sets U1 and U2, and let f ∈ F(U1 ∪ U2) be given

by g1 in U1 and g2 in U2.

Exercise 37. Show that if X is a simply connected surface, then H1(X,C) and

H1(X,Z) are trivial.

Exercise 38. Let C∗ = C\0. Consider the covering given by the open sets

U1 = C∗\R− and U2 = C∗\R+. Here R− and R+ are the negative and positive real

axis, respectively.

1. Prove that U = U1, U2 is a Leray covering for the sheaf of locally constant

functions with integer values Z.

2. Show that Z1(U ,Z) ∼= Z(U1 ∩ U2) × Z(U1 ∩ U2).

3. Show that Z(Ui) ∼= Z, for i = 1, 2.

4. Prove that the boundary operator δ : C0(U ,Z) → C1(U ,Z) is given by

(a1, a2) 7→ (a2 − a1, a2 − a1).

5. Conclude that H1(C∗,Z) ∼= Z.

Exercise 39. Show that if X is a compact surface, the mapping H1(X,Z) →H1(X,C) induced by the inclusion Z → C is injective. Here Z and C denote the

sheaves of locally constant functions with values in the integer and complex numbers

respectively.

Exercise 40. Show that the sheaf sequence:

0 → C∗ → O∗ β→ Ω → 0,

where β(f) = df/f , is exact.

Exercise 41. Show that the mapping δ∗ of 1.5.15 is well defined.

192

Exercise 42. Show that two divisors on the Riemann sphere are linearly equivalent

if and only if they have the same degree.

Exercise 43. Let g : C → C be a smooth function with compact support. Show

that g is uniformly continuous. Use this to prove that if ξ ∈ C, then the integral∫ 2π

0

g(ξ + reiθ) dθ

converges to 2πg(ξ) when r → 0 (see 2.2.1).

Exercise 44. Use the inequality |(hn+1 − hn − pn)(z)| < 2−n to prove that the

function in 2.2.2 (pg. 64) is well defined.

Exercise 45. Let X be a compact Riemann surface and f : X → C a non-constant

meromorphic function of degree d. Show that the sheaf of meromorphic functions

on X is a finite algebraic extension field of C(f) (the field of rational functions on

f) as follows:

a) Let C be the set of critical points of f ; that is, C consists of the points p ∈ X

such that dp(f) = 0. Set B = f(C) (critical values of f) and A = f−1(C). Show

that f : X\A→ C\B is a covering of degree d.

b) Let g : X → C be a non-constant meromorphic function. Let S be the set

of poles of g, S = g−1(∞). If z ∈ C\B\S and f−1(z) = p1, . . . , pd (these points

are distinct), define aj as the j-th elementary function on g(p1), . . . , g(pd). These

means that the aj , for j = 1, . . . , d satisfy the following equation:

(g(p))d + a1(f(p)) (g(p))d−1 + · · ·+ ad(f(p)) ≡ 0,

for all p in X\A\f−1(f(S)). Show that aj are holomorphic functions.

c) We want to show that aj extend to meromorphic functions on X. That will

prove that the equation displayed above holds for all points of X and will finish the

exercise. To show that, let a be a point in B ∪ f(S), choose a neighbourhood U of

a such that the poles of g on U lie in f−1(a). Let w be a holomorphic function on

U , not identically 0 but satisfying w(a) = 0. Show that (w f)n g is holomorphic

for some non-negative integer n (on f−1(U)). Let bj be the elementary symmetric

functions of (wf)n g on f−1(z) = p1, . . . , pd, for z ∈W\a, where W is an open

subset of U . Show that functions bj extend to holomorphic functions at a. Use the

relation aj = bj/wjn to show that a− j is meromorphic on X.

193

Exercise 46. Prove the Riemann-Hurwitz formula using the Riemann-Roch the-

orem.

Exercise 47. Prove the assertions of 2.4.7

Exercise 48. Let L be a line bundle on a Riemann surface X and L−1 the dual

bundle. Prove that for any point p ∈ X there is an isomorphism between (Lp)∗, the

dual space of Lp, and (L−1)p.

Exercise 49. Show that the mapping between the Picard group X and H1(X,O∗)

of 2.4.14 is a group isomorphism.

Exercise 50. Let K be a canonical divisor on a compact surface X of genus g.

Show that the dimension of H0(X,O(m =, K)) is given as in the table 1 at the end

of the exercises.

Exercise 51. Let K be a canonical divisor on a compact surface X of genus g.

Show that

dim H0(X,O(mK − p)) = dim H0(X,O(mK)) − 1,

for any point p of X.

Exercise 52. Compute the dimension of the spaces of higher order differentials.

Exercise 53. Prove that projective embeddings im (see 2.7.4 and 2.7.8) are well

defined.

Exercise 54. Recall the Weierstrass ℘ function for the torus Tτ (that is, Tτ = C/Gτ

where ℑ(τ) > 0) is given by the expression (2.6.6):

℘(z) =1

z+∑

λ6=0

(1

(z − λ)2− 1

λ2

).

Here λ is of the form λ = n +mτ with n and m integers.

a) Prove that ℘ is an even function (℘(z) = ℘(−z)).b) Show that the derivative of ℘ is given by

(℘′)(z) =∑ 1

(z − λ)3,

where the sum is taken over all points of the form λ = n + mτ , including λ = 0.

Use this expression to show that ℘′ is an odd function (℘′(−z) = −℘′(z)).

c) Let f℘ : Tτ → C be the mapping (of degree 2) induced by ℘ on the torus Tτ .

Show that the ramification points of f℘ are given by the points corresponding to 0,

194

1/2, τ/2 and (1 + τ)/2.

d) Show that ℘′ vanishes (or it has a singularity if you do not take local coordi-

nates) precisely at the four points given above (and the corresponding points on C

congruent via the group Gτ ).

e) Show that the mapping i : Tτ → P2 given by p 7→ [1 : ℘(p) : ℘′(p)] defines an

embedding of Tτ into the 2-dimensional projective space.

Exercise 55. Complete the proof of theorem 2.7.11 as follows:

a) To define the map φL : X → Pn in a neighbourhood of a point p we have used

a section σ that does not vanish at p. Show that if σ1 is another section defined in

a neighbourhood of p, say U ′, with σ1(p) 6= 0 then for every point q ∈ U ∩ U ′ we

have thatsj(q)

σ(q)= h(q)

sj(q)

σ1(q), where h is a non-zero holomorphic section. This shows

that φL does not depend on the choice of σ as long as φL(q) is in Pn (see below).

b) Use 2.6.5 to prove that the sections sj do not have common zeroes and there-

fore φL maps X into Pn.

c) In the text we have shown that if p and q are two distinct points of X then

there exists a section of L which vanishes at p but it does not vanish at q. Use this

fact to prove that φL is injective.

d) We have also shown that if p ∈ X then there exists a section of L with a

simple zero at p. Using this fact show that the differential of φL does not vanish.

This completes the proof of the fact that φL : X → Pn is an embedding.

Exercise 56. Let z1, . . . , z2g+2 be distinct points on the Riemann sphere. Prove

that the Riemann surface of the polynomial

w2 = (z − z1), . . . , (z − z2g+g)

is a compact surface X of genus g. Let z denote the function on X, and pj the

preimage of zj (j = 1, . . . , 2g + g). Without loss of generality we can assume that

the points pj are not equal to ∞ or 0. Assume the divisor of z is given by div(z) =

q3 + q4 − q1 − q2; prove that the divisor of w is given by div(w) = (p1 + · · ·+p2g+2)−(g + 1)q1 − (g + 1)q2. Use this to prove that the 1-forms zj dz

w, for j = 0, . . . , g − 1

form a basis of the space of 1-forms on X. Basis for holomorphic differentials on a

hyperelliptic surface.

195

Exercise 57. Show that if f : X → C is a meromorphic function of degree d on a

hyperelliptic surface of genus g, and d ≤ g, then d is even.

Exercise 58. Prove that if U is an open, connected subset of C, and A is a discrete

subset of U , then U\A is connected.

Exercise 59. Let 1 < α1 < · · · < αg 2g be the complementary set of the numbers

n1, . . . , ng of Weierstrass’ theorem (2.8.1) in 1, . . . , 2g. Call these numbers the

non-gaps at the point p (the nj ’s are called the gaps). Prove the following statement

regarding the non-gaps:

1. If αj and αk are two “non-gaps” with αj + αk ≤ 2g, then αj + αk is also a

“non-gap”.

2. For each integer j with 0 < j < g one has αj + αg−j ≥ 2g.

3. If α1 = 2 then αj = 2j and αj + αg−1 = 2g, for 0 < j < g.

4. If α1 > 2, then for some j with 0 < j < 2g one has αj + αg−j > 2g.

5.∑g−1

j=1 αj ≥ g(g − 1), and the equality occurs if and only if α1 = 2.

Exercise 60. Prove lemma 2.8.4 for the case of m ≥ 3.

Exercise 61. Let X be a hyperelliptic compact surface of genus g and f : X → C

a function of degree 2. Let j : X → X denote the hyperelliptic involution. This

exercise shows an explicit basis of H0(X,Ω). Assume that f is not branched over

∞; that is that the fixed points of j are not poles of f .

a) Choose a point p in X not fixed by j; show that there exists a meromorphic

function u : X → C such that u(q) 6= u(j(q)) for q in a neighbourhood of p, and

that we can choose u to be holomorphic at p and j(p).

b) It follows from exercise 45 that there is an equation of the following form:

u2(p) + 2a1(f(p)) u(p) + a2(f(p)) = 0.

Write (u+a1)2 = p/q, where p and q are polynomials on z. Then show that pq = p1q

21,

where p1 does not have multiple roots; moreover, p1(z) = c(z − z1) · · · (z − z2g+2),

where c 6= is a complex number and zj = f(pj), for pj, j = 1, . . . , 2g + 2 the

Weierstrass points of X. Show that zj 6= zk if j 6= k.

c) Define w = (u + a1)q/q1. Show that w(p1) 6= w(p2) for f−1(z) = p1, p2 in

an open set of X. Show that w = p21.

196

d) Show that w(p) = 6= w(j(p)) for p in an open set of X but w(p)2 = w(j(p))2

in that open set, and therefore on X by the identity principle. This implies that

w(p) = −w(j(p)) on X. Show that w is a function of degree g + 1 on X.

e) Define forms for j = 0, . . . , g − 1 by

ωj(p) =f j(p) f ′(p) dz

w(f(p)),

where z is a local coordinate on X. Use the identity w = p21 to show that ωj =

2f jdw/p′1 on f−1(C) and thus ωj is holomorphic at those points.

f) To prove that ωj is holomorphic show that near a pole of f we have w =

c1fg+1O(1 + 1

z) and therefore ωj = c1f

j−g−1(1 +O(1z)) is also holomorphic.

Exercise 62. Let f : X → C be a function of degree 2 on a hyperelliptic Riemann

surface X of genus g. Denote by p1, . . . , p2g+2 the Weierstrass points of X. Show

that the images of these points under f are distinct; that is, f(pj) 6= f(pk), for

1 ≤ j < k ≤ 2g + 2.

Exercise 63. Prove Noether Gap Theorem (2.8.2).

Exercise 64. If D is a divisor on a surface X we say that O(D) is globally

generated if there exists a function f ∈ H0(X,O(D)) such that ordp(f) = −D(p)

for every p inX. Equivalently, every function g defined on a neighbourhood of p with

div(g) ≥ −D can be written as g = h f , where h is holomorphic on a neighbourhood

of p. Show that if D has degree greater than 2g − 1 and X is compact of genus g

then O(D) is globally generated.

Exercise 65. Let X = C/Gτ , p0 a point of X, Dn the divisor np0. Show that

dim H0(X,O(Dn)) =

0, n < 0,

1, n = 0,

n, n > 0.

Exercise 66. Consider the homeomorphism w : C → D given by

w(z) =z

1 + |z| .

This mapping induces a Riemann surface structure on C, given by a single chart

(C, w). Denote by C1 the complex plane with this structure. Let u : C → R be

197

defined by u(z) = x1+|z| , where z = x+ iy. Show that u is harmonic on C1, but it is

not when we consider the standard structure of the complex plane.

Exercise 67. Let A be an open subset of the complex plane, p a point in the

boundary of A. Suppose that there exists a disc D centred at a point q ∈ A such

that D ∩ A p and D ⊂ A. Show that p is a regular point of A.

Hint : let c be the middle point of the segment joining p and q. Show that the

function β(z) = log(r/2) − log(|z − c|) is a barrier at p.

Exercise 68. Let X be a Riemann surface, X its universal covering space (iden-

tified with C, C or H) and G the group of deck transformations (identified with a

group of Mobius transformations). Show that X is biholomorphic to the quotient

X/G.

Exercise 69. Let A(z) = az+bcz+d

be a Mobius transformation with ad− bc = 1.

1. Show that A has order 2 (that is, A A = Id) if and only if a + d = 0.

2. Prove that A has only one fixed point in C if and only if |a+ d| = 2.

Exercise 70. The Picard subgroup G of Aut(C) is given by the transformations

of the form z 7→ az+bcz+d

, satisfying:

1. ad− bc = 1.

2. a, b, c, d are in Z[i]; that is, their real and imaginary parts are integers.

Show that G is discrete but it does not act properly discontinuously at any point of

the Riemann sphere.

Exercise 71. Prove the second lemma in 3.3.5

Exercise 72. Show that |z−w|2 − |1− wz|2| = |z|2 + |w|2 − |z|2 |w|2 − 1. Use this

to show that if |z| = 1 and |w| < 1 then | z−w1−wz | < 1. Prove proposition 3.3.8

Exercise 73. Let G be a group of Mobius transformations of the form

Tλ : z 7→ z+λ, where λ is a complex number. Let r = inf|λ; Tλ ∈ G. Show that

if there does not exist Tµ in G with r = |µ| then G cannot be discrete.

Exercise 74. Compute the area of a hyperbolic pentagon with angles given as in

picture 13. More generally, compute the area of a hyperbolic pentagon (or convex

polyhedron).

Exercise 75. Show that if X = H/G is a compact surface of genus 2 then N(G)

strictly contains G.

198

Exercise 76. Let zi, i = 1, . . . , 4 be four distinct point in the Riemann sphere.

Compute the cross ratios (zτ(1), zτ(2), : zτ(3), zτ(4)) as τ varies over all 24 permutations

of four symbols. Show that there are only 6 distinct values among all these cross

ratios.

Exercise 77. Let C be a line or circle orthogonal to S1 and A an automorphism

of the unit disc. Show that A(C) is a line or circle orthogonal to S1.

Exercise 78. Show that the mapping z 7→ −z is an isometry in the hyperbolic

metric of H.

Exercise 79. Prove lemma 3.4.4 by direct computation.

Exercise 80. Prove lemma 3.4.17.

Exercise 81. Prove that the exponential sequence

0 → Z → Cexp→ C∗ → 0

is exact.

Exercise 82. Show that the field of meromorphic functions on a compact surface

is an algebraic field of one variable. More precisely, let f : X → C be a non-constant

meromorphic function on a compact surface X; let d denote the degree of f . Let g be

another meromorphic function on X. Prove that there exist meromorphic functions

aj : C → C, j = 1, . . . , d, such that

(g(p))d + a1(f(p)) (g(p))d−1 + · · · + ad(f(p)) = 0,

for all p in X.

Exercise 83. Show that a discrete subset of the complex plane is either finite or

infinite countable.

Hint : cover C by a countable increasing sequence of compact sets.

199

g m dimension

0 m ≤ 0 1 − 2m

m > 0 0

1 0

g ≥ 1 m < 0 0

m = 0 1

m = 1 g

m > 1 (2g − 1) (g − 1)

Table 1.

φ

Figure 15. Hyperbolic pentagon.

Bibliography

[1] L. V. Ahlfors. Complex Analysis. Mc-Graw Hill, New York, USA, 1966.

[2] L. V. Ahlfors and L. Sario. Riemann surfaces. Princeton University Press, Princeton, New

Jersey, USA, 1960.

[3] W. M. Boothby. Introduction to Differential Manifolds and Riemannian Geometry. Pure and

Applied Mathematics. Academic Press, New York, 1975.

[4] G. E. Bredon. Topology and Geometry, volume 139 of Graduate Texts In Mathematics.

Springer-Verlag, Berlin and New York, 1993.

[5] J. B. Conway. Functions of one complex variable, volume 11 of Graduate Texts In Mathematics.


[6] J. B. Conway. A Course in Functional Analysis, volume 96 of Graduate Texts In Mathematics.


[7] H. Farkas and I. Kra. Riemann Surfaces, volume 72 of Graduate Text in Mathematics.

Springer-Verlag, New York, Heidelberg and Berlin, 2nd edition, 1992.

[8] O. Forster. Lectures on Riemann Surfaces, volume 81 of Graduate Text in Mathematics.

Springer-Verlag, New York, Heidelberg and Berlin, 1981.

[9] M. J. Greenberg and J. R. Harper. Algebraic Topology, A First Course. The Benjamin Cum-

mings Publising Company, Reading, Massachusetts, U.S.A., 1981.

[10] P. Griffths and J. Harris. Principles of Algebraic Geometry. Wiley Classics Library. John

Wiley & Sons, New York, USA, 1978. (First edition 1951).

[11] N. Jacobson. Basic Algebra, volume I. Hindustani Publishing Corporation (India), New Dehli,

India, 1974. Originally published by W. H. Freedman, 1974.

[12] S. Lang. Algebra. Addinson-Wesley Publishing Company, Reading, Massachusetts, U. S. A.,

3rd edition, 1993.

[13] W. S. Massey. Algebraic topology: an introduction. Harcourt, Brace & World, New York, 1967.

[14] W. S. Massey. Singular Homology Theory, volume 70 of Graduate Text in Mathematics.

Springer-Verlag, New York, 1980.

[15] J. Milnor. Topology from a Differentiable Point of View. Universiry Press of Virginia, Char-

lottesville, U.S.A., 1965.

[16] J. R. Munkres. Topology: a first course. Prentice-Hall of India, New Delhi, 11th indian edition,

1975.

201

202

[17] J. R. Munkres. Topology: a first course. Prentice-Hall, Englewood Cliffs, 1975.

[18] S. Nag. The Complex Analytic Theory of Teichmuller Spaces. John Wiley & Sons, 1988.

[19] R. Narasimhan. Compact Riemann Surfaces. Lectures in Mathematics, ETH Zurich.

Birkhauser, Basel-Boston-Berlin, 1992.

[20] W. Rudin. Real and Complex Analysis. McGraw-Hill, New York, USA, 1966.

[21] W. Rudin. Functional Analysis. McGraw-Hill, New York, USA, 1973.

[22] W. Rudin. Real and Complex Analysis. McGraw-Hill, New York, USA, international Edition

edition, 1987.

[23] R. R. Simha. The Uniformisation Ttheorem for planar Riemann surfaces. Arch. Math., 53:599–

603, 1989.

[24] R.R. Simha. The Riemann-Roch theorem for compact Riemann surfaces. Ensign. Math. (2),

27:185–196, 1981.

[25] M. Spivak. A Comprehensive Introduction to Differential Geometry, volume 1. Publish or

Perish, Houston, USA, second edition, 1971.

203

Notation

Term Page Meaning

Z - Integers

R - Real numbers

C - Complex numbers

C∗ - z ∈ C; z 6= 0A\B - p ∈ A; p /∈ BIdX - Identity function on the set X

A ⊔ B - disjoint union of sets A and B

∂U - boundary of the set U

D 4 unit disc, z ∈ C; |z| < 1fx, fy, fz, fz 2 partial derivatives

c1 ∼ c2 8 homotopic paths

Ω(X, x0) 9 loops on the manifold X based at x0

π1(X, x0) 9 fundamental group

Deck(X/Y ) 10 group of deck transformations (of a covering X → Y )

supp(f) 11 support of a function

|| · || 11 norm

S2 13 2-sphere, (x, y, z) ∈ R3; x2 + y2 + z2 = 1Cn(X) 13 chain group of X

δ : Cn(X) → Cn−1(X) 14 boundary operator between chain groups

Hn(X,Z) 14 homology groups

χ(X) 15 Euler-Poincare characteristic

H 19 upper half plane, z ∈ C; Im(z) > 0C 19 Riemann sphere or extended complex plane, C ∪ ∞Tτ 21 torus

bp(f) 24 branching number of the function f at the point p

B(f) 27 total branching number of the function f ,∑

p∈X bp(f)

C(X) 29 sheaf of continuous functions on X

204

S(X) 29 sheaf of smooth functions on X

S1(X) 30 sheaf of smooth 1-forms on X

S(1,0)(X) 30 sheaf of (1, 0) type forms on X

S(0,1)(X) 30 sheaf of (0, 1) type forms on X

S2(X) 31 sheaf of smooth 2-forms on X

ω1 ∧ ω2 31 exterior product of forms

d 32 exterior derivative

∂ 33 delta operator

∂ 33 delta bar operator

∗ 33 conjugation operator

T ∗X 36 cotangent space of a manifold X

M1(X) 40 meromorphic 1-forms on X

F 42 sheaf (or presheaf) on X

O(X) 43 sheaf of holomorphic functions on the surface X

M(X) 43 sheaf of meromorphic functions on the surface X

O∗(X) 43 sheaf of non-zero holomorphic functions on the surface X

M∗(X) 43 sheaf of meromorphic, non-identically zero functions

on the surface X

Cn(U ,F) 44 cochain group

Z1(U ,F) 44 group of cocycles

B1(U ,F) 44 group of coboundaries

H1(U ,F) 44 cohomology group with respect to UH1(X,F) 47 1st cohomology group

|F| 50 sheaf associated to a presheaf

α : F → G 50 sheaf homomorphism

K 51 kernel sheaf

S1c 54 sheaf of closed forms on X

H1dR(X) 55 de Rham cohomology group

K(U) 44 kernel sheaf

D 59 divisor on a surface

D ≥ 0 59 effective divisor

205

div(f) 59 divisor of a function

div(ω) 60 divisor of a form

D1 ∼ D2 60 linearly equivalent divisors

deg (D) 60 degree of divisor D

DivP (X) 61 group of principal divisors of X

KX 61 canonical class of X

O(D) 61 sheaf of meromorphic functions associated to a divisor

Ω(D) 62 sheaf of meromorphic forms associated to a divisor

∆ 65, 128 Laplacian operator

Cx 68 skyscraper sheaf

L⊗ L′ 75 tensor product of bundles

L−1 75 dual bundle of L

Pic(X) 77 Picard group of the surface X

L(D) 79 line bundle associated to the divisor D

sD 80 canonical section of the bundle L(D)

deg (L) 81 degree of the bundle L

KX 86 canonical line bundle of the surface X

det(V ) 88 determinant line bundle of the bundle V

Res 93 Residue isomorphism between H1(X,KX) and C

℘ 97 Weiertrass ℘-function

Pn 100 projective space

i1 : X → Pg−1 101 canonical mapping

W (f1, . . . , fm) 109 Wronskian of the functions f1, . . . , fm

Aj(ψ), Bj(ψ) 116 periods of the form ψ

J(X) 121 Jacobian variety of the surface X

PSL(2,C) 150 projective special linear group

Ω(G) 158 region of discontinuity of the group G

(z1, z2; z3, z4) 163 cross ration of four complex numbers

Dp(G) 173 Dirichlet region of G relative to p

N(G) 176 normalizer of G (in Aut(H))

A(z0, r1, r2) 179 annulus of centre z0 and radii r1 and r2

206 Index

Index

Numbers written in italic refer to the page where the corresponding entry is

described; numbers underlined refer to the definition; numbers in roman refer to the

pages where the entry is used.

Symbols

1-form . . . . . . . . . 30, 37

1-formintegration . . . . 38

1-forminvariance prop-

erty . . . . . . . . . 30

1-formmeromorphic . . . 40

1-formperiods . . . . . . 116

1-formsheaf of . . . . . . . 30

1-formtype (0, 1) . . . . . 30

1-formtype (1, 0) . . . . . 30

2-form . . . . . . . . . 31, 37

2-formintegration . . . . 39

2-forminvariance prop-

erty . . . . . . . . . 31

fz, fz . . . . . . . . . . . 2, 30

2-sphere . . . . . . . . . . . 13

2-sphereEuler-Poincare

characteristic . . 15

A

Abel’s theorem . . 122, 123

Abel-Jacobi map . . . . 121

ample line bundle . . . 106

annulus theorem . . . . 142

arithmetic genus . . . . . 68

automorphism 153, 155, 156

B

Banach space . . . . . . . 12

Banach spacecompact

mapping . . . . . . 12

barrier . . . . . . . . . . . 140

Betti number . . . . . . . 15

biholomorphism . . . . . 23

boundary operator . . . 14

branched covering . 10, 21

branching number . 24, 27

bundle . . . . . . . . . see

also vector bundle

C

canonical class . . . . . . 61

canonical classdegree . . 89

canonical divisor . . . . . 61

canonical line bun-

dle . . . . . . . 86, 106

canonical mapping . . 101

canonical section . . . . . 80

Cauchy-Riemann equa-

tions . . . . . . . . . 3

chain group . . . . . . . . 13

changes of coordinates . 18

closed form . . . . . . . . . 33

co-closed form . . . . . . . 33

co-exact form . . . . . . . 33

coboundary . . . . . 14, 44

coboundary operator . . 44

coboundarysplits . . . . . 44

cochain group . . . . . . . 44

cocycle . . . . . . . . . . . . 44

cohomology group . . . .

44, 47, 49, 56

cohomology groupexact

sequence . . . . 53, 56

cohomology groupmero-

morphic functions 96

cohomology groupof a

compact surface 66


disc . . . . . . . 65, 86


line bundle . . 78, 80


vector bundle . . 82

cohomology groupof the

complex plane 65, 85

cohomology groupRie-

mann sphere . . . 66

Index 207

cohomology groupsmooth

functions . . . . . 84

commutator subgroup . 14

compact mapping . . . . 12

compact surfacetriangu-

lation . . . . . . . . 27

complex atlas . . . . . . . 18

complex atlasequiva-

lence . . . . . . . . 18

complex atlasmaximal . 19

complex manifold . . . . 72

complex planeautomor-

phisms . . . . . . . 153

complex planecohomol-

ogy group . . . 65, 85

complex structure . . . . 18

conjugation . . . . . . . . 33

connecting homomor-

phism . . . . . . . . 52

continuous functionsheaf

of . . . . . . . . . . . 29

coordinate patch . . . . . 18

cotangent space . . 36–37

countable basis of topol-

ogy . . . . . . . 7, 19

covering . . . . . . . . . . . . 7

covering map . . . . . . . . 7

covering transformation 10

coveringbranched . . . . 10

coveringuniversal . . . . . 9

cross ratio . . . . . 163, 164

curve . . . . . see also path

cycle . . . . . . . . . . . . . 14

cylinder . . . . . . . . . . 154

D

de Rhamcohomology . . 55

de Rhamgroup . . . . . . 55

de Rhamsequence . . . . 54

de Rhamtheorem . . . . 55

deck transformation . . 10

degree . . . . . . . . . 24, 60

degreeof a divisor . . . . 60

degreeof a line bundle . 81

determinant line bundle 88

differential of a smooth

mapping . . . . . . 101

Dirichlet region . . 173, 174

divisor . . . . . . . . . 59, 62

divisor class group . . . 61

divisor of a meromorphic

form . . . . . . . . . 60

divisor of a meromorphic

function . . . . . . 59

divisorassociated line

bundle . . . . . . . 79

divisorassociated sheaf 61

divisorcanonical . . . . . 61

divisordegree . . . . . . . 60

divisoreffective . . . . . . 59

divisorglobally gener-

ated . . . . . . . . . 196

divisorgroup of . . . . . . 59

divisorlinear equiva-

lence . . . . . . 60, 80

divisorof a meromorphic

form . . . . . . . . . 60

divisorof a meromorphic

function . . . . . . 59

divisorprincipal . . . . . . 61

divisorRiemann-Roch

theorem . . . . 69, 95

Dolbeault’s lemma . . . 64

Dolbeault’s lemmacom-

pact support case 62

dual bundle . . . . . . . . 75

E

effective divisor . . . . . . 59

elliptic Mobius transfor-

mation . . . 152, 158

embedding . 101, 104, 105

essential singularity . . . . 4

Euler-Poincare charac-

teristic . . . . . 15, 16

evenly covered neigh-

bourhood . . . . . . 7

exact form . . . . . . . . . 33

exact sequence . . . . . . 51

exact sequence in coho-

mology . . . . . 53, 56

exterior algebra . . . . . 32

exterior derivative . . . . 32

exterior product . . . . . 31

F

form . . . . . . . . . 30, 31, 37

formclosed . . . . . . . . . 33

formco-closed . . . . . . . 33

208 Index

formco-exact . . . . . . . . 33

formconjugation . . . . . 33

formexact . . . . . . . . . . 33

formexterior derivative 32

formexterior product . . 31

formharmonic . . . . . . . 34

formholomorphic . . . . . 34

formmeromorphic . . . . 40

formperiods . . . . . . . 116

formresidue . . . . . . . . 41

formsupport . . . . . . . . 39

Fuchsian group . . . . . 170

fundamental domain . 172

fundamental do-

mainDirichlet re-

gion . . . . . 173, 174

fundamental group . 9, 13

Fundamental Theorem of

Algebra . . . . . . 26

G

Gap theorem . . . 107, 108

Gauss-Bonet theo-

rem . . . . . 169, 175

general linear group . 150

genus . . . . . . . . . . 12, 68

genusarithmetic . . . . . 68

genusarithmetic and

topological are

equal . . . . . . . . 91

genustopological . . . . . 12

geodesic . . . . . . . 167, 168

globally generated divi-

sor . . . . . . . . . . 196

group of divisors . . . . . 59

H

harmonic form . . . . . . 34

harmonic function 34, 128

harmonic functionbar-

rier . . . . . . . . . 140

harmonic functionHar-

nack’s inequality 134

harmonic functionHar-

nack’s principle . 134

harmonic functionMax-

imum Modulus

Principle . . . . . 132

harmonic function-

mean value prop-

erty . . . . . 131, 133

harmonic functionon a

Riemann surface 138

harmonic functionPer-

ron’s method . . 139

Harnack’s inequality . 134

Harnack’s principle . . 134

holomorphic form . . . . 34

holomorphic function 2, 22

holomorphic function-

local representa-

tion . . . . . . . . 3, 6

holomorphic func-

tionorder of a zero 3

holomorphic function-

power series . . . . 3

holomorphic functionsev-

eral variables . . 72

holomorphic functionuni-

form convergence 6

holomorphic mapping . 22

holomorphic mappingde-

gree . . . . . . . . . 24

holomorphic mappingex-

amples . . . . . . . 23

holomorphic mappinglo-

cal representation 23

holomorphic mappingRiemann-

Hurwitz relation 27

holomorphic section . . 76

holomorphic vector bun-

dle . . . . . . . . . . 72

homology group . . 14, 15

homology groupBetti

number . . . . . . 15

homotopy . . . . . . . . . . . 8

Hurwitz theorem . . . . 177

hyperbolic distance . . 163

hyperbolic distancecom-

pleteness . . . . . 169

hyperbolic distancein-

duced topology . 166

hyperbolic metric 163, 165

hyperbolic metricGauss-

Bonet theorem . 169

hyperbolic metric-

geodesic . . . . . . 168

hyperbolic metricisome-

tries . . . . . . . . . 166

Index 209

hyperbolic Mobius trans-

formation . . . . . 152

hyperelliptic involution 114

hyperelliptic Riemann

surface . . . . . . .

102, 112, 113, 115

hyperelliptic Riemann

surfacehyperellip-

tic involution . . 114

I

Identity Principle . . 4, 23

integration of forms 38–39

integration of 2-forms . 39

isolated singularity . . . . 4

isolated singularityessen-

tial singularity . . 4

isolated singularity-

pole . . . . . . . . 4, 5

isolated singularityre-

movable singular-

ity . . . . . . . . . 4, 5

isomorphic vector bun-

dles . . . . . . . . . 75

isomorphism of bundles 74

J

Jacobi inversion prob-

lem . . . . . . . . . 124

Jacobian variety . . . . 121

K

kernel sheaf . . . . . . . . 51

Kleinian group . . 158, 159

Kleinian groupfunda-

mental domain . 172

Kleinian groupregion of

discontinuity . . . 158

Koebe theorem . . . . . 141

L

Laplacian . . . . . . . . . . 34

Leray’s theorem . . . . . 48

lift of a path . . . . . . . . . 8

line bundle . . . . . . 73, 88

line bundleample . . . . 106

line bundleassociated to a

divisor . . . . . . . 79

line bundlecanonical 86, 106

line bundlecanonical sec-

tion . . . . . . . . . 80

line bundlecohomology

group . . . . . . 78, 80

line bundledegree . . . . 81

line bundlemeromorphic

section . . . . . . . 78

line bundletrivial . . . . 83

line bundlevery ample 106

linearly equivalent divi-

sors . . . . . . . 60, 80

local coordinates . . . . . 18

local trivialization . . . . 73

locally path connected

space . . . . . . . . . 8

loop . . . . . . . . . . . . . . . 9

loxodromic Mobius

transformation . 152

M

manifold . . . . . . . . . . . . 7

manifoldcomplex . . . . . 72

manifoldsmooth . . . . . 28

manifoldsurface . . . . . . . 7

mappingdifferential . . 101

maximal complex atlas 19

Maximum Modules The-

orem . . . . . . . . . 5

Maximum Modulus Prin-

ciple . . . . . . 23, 132

mean value prop-

erty . . . . . 131, 133

meromorphic form . . . 40

meromorphic formdivi-

sor . . . . . . . . . . 60

meromorphic function 5, 22

meromorphic functionco-

homology . . . . . 96

meromorphic functiondi-

visor . . . . . . . . 59

meromorphic functionex-

istence . . . . . 71, 79

meromorphic section 77, 78

meromorphic sectionexis-

tence . . . . . . . . 78

Montel theorem . . . . . . 7

morphism of bundles . . 74

multiplicity . . . . . . . . 110

Mobius transformation 26

Mobius transformation-

commuting . . . . 160

210 Index

Mobius transformationel-

liptic . . . . 152, 158


hyperbolic . . . . 152


loxodromic . . . . 152


parabolic . . . . . 152

N

Noether Gap theorem 108

normal family . . . . 6, 141

normalizerof a Fuchsian

group . . . . . . . . 176

normalizerof a subgroup 176

O

Open Mapping Theo-

rem . . . . . . . 5, 23

P

parabolic Mobius trans-

formation . . . . . 152

partition of unity . . . . 11

path . . . . . . . . . . . . . . . 8

path connected space . . 8

pathhomotopy . . . . . . . 8

pathlift . . . . . . . . . . . . . 8

pathloop . . . . . . . . . . . . 9

pathpiecewise smooth . 38

periods of a form . . . 116

Perron’s method . . . . 139

Picard group . . . . . . . 77

piecewise smooth path 38

planar Riemann sur-

face . . . . . 145, 147

Poincare lemma . . 33, 54

Poincare-Koebe Uni-

formization theo-

rem . . . . . . . . . 147

Poisson kernel . . . 129, 131

pole . . . . . . . . . . 4, 5, 22

poleorder of . . . . . . . . . 5

poleresidue . . . . . . . . . . 5

power series . . . . . . . . . 3

presheaf . . . . . . . . . . . 42

presheafassociated sheaf 49

presheafrestriction homo-

morphisms . . . . 42

presheafstalk . . . . . . . 50

principal divisor . . . . . 61

projective general linear

group . . . . . . . . 150

projective space . . . . 100

projective special linear

group . . . . 150, 157

properly discontinuous

action . . . . . . . . 158

punctured plane . . . . 154

R

ramification number . . 24

rank of a vector bundle 72

rational function . . . . . 26

Reciprocity theorem . 120

refinement mapping . . 45

region of discontinuity 158

removable singularity 4, 5

Removable Singularity

Theorem . . . . . . 5

residue . . . . . . . . . . 5, 41

Residues theorem . . . . 41

restriction homomor-

phisms of a

presheaf . . . . . . 42

Riemann mapping theo-

rem . . . . . . . . . 141

Riemann sphere . . 19, 71

Riemann sphereunique

complex structure 71

Riemann surface . . 18, 72

Riemann surfaceabelian

fundamental

group . . . . . . . . 161

Riemann surface-

branched covering 21

Riemann surfacecomplex

plane . . . . . . . . 19

Riemann surfacecomplex

structure . . . . . 18

Riemann surfacecount-

able basis of topol-

ogy . . . . . . . . . 19

Riemann surfacecylin-

der . . . . . . . . . . 154

Riemann surfaceexam-

ples . . . . . . . 19–22

Riemann surfaceGauss-

Bonet theorem . 175

Index 211

Riemann surfacehar-

monic function . 138

Riemann surfaceholo-

morphic mapping 22

Riemann surfaceHurwitz

theorem . . . . . . 177

Riemann surfacehyperel-

liptic . . . . . . . .

102, 112, 113, 115

Riemann surfacelocal co-

ordinates . . . . . 18

Riemann surfacepla-

nar . . . . . . 145, 147

Riemann surfacepunc-

tured plane . . . . 154

Riemann surfaceRiemann-

Roch theorem . . 87

Riemann surfaceSerre

Duality theo-

rem . . . . . . . 94, 95

Riemann surfacesubhar-

monic function . 138

Riemann surfacetorus 154

Riemann surfaceunit

disc . . . . . . . 4, 19

Riemann surfaceupper

half plane . . . . . 19

Riemann’s Bilinear Rela-

tions . . . . . . . . 119

Riemann-Hurwitz rela-

tion . . . . . . . . . 27

Riemann-Roch theo-

rem . . . . 69, 87, 95

rotation . . . . . . . . . . 155

S

Sard’s theorem . . . . . . 40

Schwartz’s theorem . . . 68

Schwarz lemma . . . 4, 155

section . . . . . . . . . . . . 76

sectioncanonical . . . . . 80

sectionholomorphic . . . 76

sectionmeromorphic . . 77

Serre Duality theo-

rem . . . . . . . 94, 95

sheaf . . . . . . . . . . . . . 43

sheafassociated to a divi-

sor . . . . . . . . . . 61

sheafassociated to a

presheaf . . . . . . 49

sheafcoboundary opera-

tor . . . . . . . . . . 44

sheafcochain group . . . 44

sheafcohomology group

44, 47, 49, 56

sheafconnecting homo-

morphism . . . . . 52

sheafde Rhamsequence 54

sheafexact sequence . . . 51

sheafhomomorphism . . 50

sheafkernel sheaf . . . . . 51

sheafLeray’s theorem . . 48

sheafmeromorphic func-

tionscohomology 96

sheafof continuous func-

tions . . . . . . . . 29

sheafof smooth func-

tions . . . 29, 47, 84

sheafof 1-forms . . . . . . 30

sheafof 2-forms . . . . . . 31

sheafskyscraper . . . . . . 68

sheafstalk . . . . . . . . . . 50

Shwartz’s theorem . . . . 12

side derivative . . . . . . . 38

simplex . . . . . . . . . . . 13

simply connected space . 9

singularity . . . see also

isolated singularity

skyscraper sheaf . . . . . 68

smooth function . . . . . 29

smooth functiondifferen-

tial . . . . . . . . 32, 33

smooth functionexterior

derivative . . . . . 32

smooth functionsheaf

of . . . . . . . . . 29, 47

smooth manifold . . 28, 40

smooth manifoldclassifi-

cation . . . . . . . . 40

smooth mappingdifferen-

tial . . . . . . . . . . 101

special linear group . . 150

stalk . . . . . . . . . . . . . 50

Stokes’ theorem . . . . . 39

subharmonic func-

tion . . . . . 135, 138

subharmonic functionon

a Riemann sur-

face . . . . . . . . . 138

212 Index

subharmonic function-

Perron’s method 139

support of a 2-form . . . 39

surface . . . . . . . . . . . . . 7

surfacefundamental

group . . . . . . . . 13

surfacehomology

group . . . . . . 14, 15

T

tangent space . . . . . . . 37

tensor product . . . . . . 75

torus . . . . . . 20, 124, 154

torusWeierstrass ℘-

function . . . . . . 97

total branching number 27

transition functions . . . 74

triangulation . . . . 15, 27

trivial bundle . . . . 75, 76

U

Uniformization theo-

rem . 147, 150, 158

unit disc . . . . . . . . . . . . 4

unit discautomorphisms 155

universal covering space 9

upper half planeautomor-

phisms . . . . . . . 156

V

vector bundle . . . . . . . 72

vector bundle isomor-

phism . . . . . . . . 74

vector bundle morphism 74

vector bundledual . . . . 75

vector bundleisomorphic 75

vector bundleisomor-

phism . . . . . . . . 74

vector bundleline bundle 73

vector bundlelineample 106

vector bundlelinevery

ample . . . . . . . . 106

vector bundlelocal trivi-

alization . . . . . . 73

vector bundlemeromor-

phic section . . . 77

vector bundlemorphism 74

vector bundlePicard

group . . . . . . . . 77

vector bundlerank . . . . 72

vector bundlesection . . 76

vector bundletensor

product . . . . . . 75

vector bundletransition

functions . . . . . 74

vector bundletriv-

ial . . . . . 73, 75, 76

very ample line bundle 106

W

Weierstrass Gap theo-

rem . . . . . . . . . 107

Weierstrass point 108,

109, 112, 113, 115

Weierstrass theorem . . . 6

Weierstrass ℘-function . 97

Wronskian . . . . . . . . 109

List of Figures

1 Homotopy between two paths. 9

2 A surface of genus 2 with generators of the fundamental group. 13

3 Allowed and wrong triangulations. 16

4 Condition (3) in definition 1.2.7. 16

5 A triangulation of the sphere. 17

6 Branched covering. 21

7 Subdivision of triangles (z0 is a critical value). 27

8 Proof of theorem 1.4.26. 42

9 Proposition 1.5.12. 48

10 Polygon ∆ corresponding to a surface of genus 2. 116

11 Proof of lemma 2.9.2. 118

12 h−1(t). 143

13 Triangle with two zero angles. 170

14 Gauss-Bonet. 170

15 Hyperbolic pentagon. 199

213

riemann surfaces

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