complexiii

Complex Manifolds by Will MerryLecture notes based on the Complex Manifolds course lectured by Dr. A.G. Kovalev inLent term 2008 for Part III of the Cambridge Mathematical Tripos.

Contents

1 Introduction to complex manifolds . . . . . . . . . . . . . . . . . . . . . 12 Differential (p, q)-forms and Dolbeault cohomology . . . . . . . . . . . . 93 Almost complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 154 Holomorphic vector bundles, subvarieties and some commutative algebra 195 Meromorphic functions and divisors . . . . . . . . . . . . . . . . . . . . 246 The Chern connection and the first Chern class . . . . . . . . . . . . . . 287 Blow ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 Hermitian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 The Hodge theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010 Compact Khler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 56

Please let me know on [email protected] if you find any errors - Im sure there are many!

1 Introduction to complex manifolds

1.1 Recap on holomorphic functions of one complex variableRecall that a function f : U C C is called holomorphic if any of the following threeequivalent conditions hold:

1. f is complex analytic: at any point a U there exists (cn)n0 C such that on D(a, r) :={|z a| < r} we have f(z) = n=0 cn(z a)n.

2. f is complex differentiable: that if, f satisfies theCauchy-Riemann equations. Writingz = x+ iy and

z:=

12

(

x i

y

),

z:=

12

(

x+ i

y

), (1)

we havef

z= 0.

3. f satisfies Cauchys integral formula: given a disc D U and z D we have

f(z) =1

2pii

D

f(w)w z dw.

1.2 Holomorphic functions of several complex variablesA polydisc is is an open set of the form

P (a, r) :={z Cn | |zi ai| < ri} = D (a1, r1) D (an, rn) ,

where each D(ai, ri) C is a standard open disc.

1

1 Introduction to complex manifolds 2

Note that the polydiscs form a basis for the Euclidean topology on Cn, and thus there will beno loss of generality if we consider functions defined on polydiscs where convenient. In these notes,D will denote a generic disc and P = D1 Dn a generic polydisc. A C1 function f : P Cnis called holomorphic if for i = 1, . . . , n, the function

gi : Di C, w 7 f(z1, . . . , zi1, w, zi+1, . . . , zn

)is a holomorphic function of one complex variable w. In other words, f is holomorphic if and onlyif

f

zi= 0 for i = 1, . . . , n.

1.3 Theorem (complex analyticity of holomorphic functions)Let U Cn be open and f : U C be C1. Then f is holomorphic if and only if it is complexanalytic, that is, for any a U there exists a polydisc P (a, r) U (the notation P U impliesP U and P is compact) such that for all z P (a, r) we have

f(z) =

i1,...,in=0

1i1! . . . in!

i1++in

(z1)i1 . . . (zn)inf(a)

(z1 a1)i1 . . . (zn an)in .

1.4 Theorem (Cauchys integral formula for several complex variables)Let U Cn be open, and P U . Let f : U C be C1. Then f is holomorphic if and only iffor any z U , we have

f(z) =1

(2pii)n

D1

. . . . . .

Dn

f(w)(w1 z1) . . . (wn zn)dw

1 dwn.

Note that the integral is not over the boundary of a polydisc (unless n = 1), but rather over ann-dimensional submanifold.

1.5 DefinitionA function : U Cm Cn is holomorphic if each coordinate function i : U C is holomor-phic.

1.6 PropositionLet : U Cm Cn be holomorphic and bijective onto its image. Then 1 is holomorphic,and so is biholomorphic.

1.7 DefinitionLet : U Cm Cn be smooth. Let (z1, . . . , zm) be coordinates on U and (w1, . . . , wn) becoordinates on Cn. Write zi = xi + iyi and wi = ri + isi, and write =

(1, . . . ,n

), where

i = ui + ivi. The complex Jacobian J(z) at z U is the linear map

J(z) :=[i

zj(z)]

: Cm Cn.

If J(z) is surjective then z is said to be a regular point for . If w Cn is such that for allz 1(w), z a regular point of , then w is said to be a regular value of .

The smooth map : U Cm = R2m Cn = R2m defines an R-linear map d(z) : Tz(R2m

)T(z)

(R2n

). With respect to the bases{

x1z, . . . ,

xmz,

y1z, . . . ,

ymz

}


of Tz(R2m

)and {

r1(z)

, . . . ,

rn(z)

,

s1(z)

, . . . ,

sn(z)

}of T(z)

(R2n

), d(z) is given by the real Jacobian matrix JR(z); namely the 2n 2m matrix

JR(z) :

[ uixj (z)] [uiyj (z)][vi

xj (z)] [

vi

yj (z)] : R2m R2n. (2)

1.8 Relating J(z) and JR(z)Let TC,z

(R2m

):= Tz

(R2m

)R C. We may complex linearly extend d(z) to define a mapd(z) : Tz

(R2m

)C T(z) (R2n) .The complex linear extension of d(z) is still given by JR(z); only now instead of an element ofMat(2n 2m,R) it is thought of as an element of Mat (2n 2m,C). Easy linear algebra showsthat {

z1z, . . . ,

zmz,

z1z, . . . ,

zmz

}(where zi and

zi are defined by (1)) forms a basis of TC,z

(R2m

); similarly{

w1(z)

, . . . ,

wn(z)

,

w1(z)

, . . . ,

wn(z)

}forms a basis of TC,(z)

(R2n

). With respect to this basis, JR(z) has matrix [izj (z)] [izj (z)][

i

zj (z)] [

i

zj (z)] ,

where i = ui ivi. In particular, if is holomorphic (so izj = i

zj 0 for all i, j) then

JR(z) =[J(z) 0

0 J(z)

].

In particular, if m = n then we have shown:

1.9 LemmaLet : U Cn Cn be holomorphic. Then for z U ,

det JR(z) = |det J(z)|2 0.

1.10 Theorem (complex inverse function theorem)Let : U V be holomorphic, where U, V Cn. Let z U be a regular point for . Then thereexists a neighborhood U0 of z such that if V0 = (U0) then : U0 V0 is biholomorphic.

1.11 Theorem (complex implicit function theorem)Let U Cm be open, and : U Cn where m n. Let z0 U be such that (z0) = 0, andsuppose that there exist distinct j1, . . . , jn {1, . . . ,m} such that

det[i

zjk

]k=1,...,n

6= 0.

Then there exist opens sets U1 Cmn and U2 Cn such that U1 U2 U , and a uniqueholomorphic function : U1 U2 such that

1(0) (U1 U2) = {(z, (z)) | z U1}.


1.12 DefinitionsA complex manifold Xn is a second countable Hausdorff space equipped with a holomorphicatlas, that is, a collection {(U, h) | A} such that X =

U and each h : U h(U)

Cn is a homeomorphism such that h h : h(U U) h(U U) is holomorphic wheredefined. Note that a complex n-manifold is also a real smooth 2n-manifold with a holomorphicatlas; we will often denote this smooth manifold by XR. It is possible to have to non-biholomorphiccomplex manifolds whose underlying real smooth manifolds are diffeomorphic.

A map : Xm Y n between complex manifolds with holomorphic atlases {(U, h)} and{(V , k)} is holomorphic if the maps k h1 are holomorphic where defined.

We let OX (we will omit the X where possible) denote the sheaf of holomorphic functionson X, that is,

OX(U) := {holomorphic functions f : U X C} .We let Op denote the local ring of germs of holomorphic functions at p X.

1.13 Proposition (Liouvilles Theorem)Let X be a compact connected complex manifold and f O(X) holomorphic. Then f is constant,and so O(X) = C.

J Since X is compact and f continuous, Re(f) is continuous and bounded, and so attains itsmaximum at some p X. Let (U, h) be a chart about p, and let u := Re(f)h1 : h(U) R2n R.Then u is harmonic, since zi (f h1) = 0 for each i. But then u is a harmonic function definedon an open set of R2n that attains its maximum in the interior of its domain. The strong maximumprinciple implies that u is constant.

Thus for any coordianate domain V such that V U 6= , we have Re(f) constant on V . SinceX is connected, we conclude that Re(f) is constant. A similar argument shows that Im(f) isconstant, and the proof is complete. I

1.14 Projective manifoldsThere is no result corresponding to Whitneys Embedding Theorem in the complex category. Infact, a compact complex manifold X never has a holomorphic embedding in CN for any N N.Indeed, suppose i : X CN is holomorphic. Let (z1, . . . , zn) be local coordinates on i (X). Thenthe map zi i : X C is a non constant holomorphic function, unless i(X) = pt. LiouvillesTheorem completes the proof. A projective manifold is a manifold that admits an embeddinginto CPN for some N N.

1.15 Examples1. A Riemann surface is a 1-dimensional complex manifold. Riemann surfaces are classified

by the Uniformisation Theorem, which asserts that all Riemann surfaces are quotientsof C, S2 = CP 1 or the unit disc D. More precisely, the Riemann surfaces are CP 1, C, acylinder C/Z , an elliptic curve C/, where = Z + Z , Im() > 0 or D/, where isa subgroup of the Mobius transformations acting properly discontinuously on D. It can beshown that all compact Riemann surfaces are projective.

2. Let V be an n-dimensional complex vector space, and let Gr(k, V ) denote the set of all k-dimensional complex linear subspaces of V . Gr(k, V ) is a complex Grasmmanian, andis a compact complex manifold of dimension k(n k). Taking k = 1 and V = Cn recoversthe complex projective space CPn. It can be shown that all complex Grassmanians areprojective, via the Plucker embedding.

3. Let be a discrete additive subgroup of Cn isomorphic to Z2n (a lattice). A complex torusis the space Cn/, given the quotient topology and the holomorphic atlas defined as follows:if pi : Cn Cn/ is the quotient map, then for any z Cn, there exists a neighborhoodU of z on which pi is invertible, and we let h = (pi|U )1 pi(U) U be a chart about pi(z).


Then any two overlapping charts have transition function of the form z 7 z + (z) where : V Cn is continuous and hence constant, and thus the transition functions areholomorphic. All complex tori Cn/ are diffeomorphic to (S1)2n as real smooth manifolds,but they are not necessarily biholomorphic. In fact, when n = 1 (the elliptic curves) thereare already uncountably many nonbiholomorphic such tori.

4. Let Hn = (Cn\{0}) /z 2z. H is called the Hopf manifold, and is an n-dimensionalcomplex manifold homeomorphic to S2n1S1. It is not immediately clear why Hn admitsthe structure of a complex manifold; we will deduce this shortly (see Corollary 1.17). It canbe shown, although we will not be in a position to do so until almost at the end of the course(see Corollary 9.20), that the Hopf manifold Hn is not projective.

The following proposition generalises Example 3 above.

1.16 Proposition (quotient manifolds)Let Xn be a complex manifold and a subgroup of biholomorphic maps of X onto itself, andsuppose that for each compact K X the set {g | g(K) K 6= } is finite. Suppose furtherthat for g 6= idX has no fixed points. Then the orbit space X/ = {(p) | p X} admits aholomorphic atlas making X/ into a complex manifold such that the quotient map pi : X X/is holomorphic.

J Firstly, the map pi is an open map, since if U X is open then

pi1 (pi(U)) =g

g(U),

which is open. It is then trivial that X/ is second countable: indeed, if {Un} is a countablebasis for the topology of X then {pi(Un)} is a countable basis for the topology of X/, since X/carries the quotient topology and pi is an open map. Given p X, pick a chart (U, h) about p andcompact set Kp,n containing p such that h (Kn,p) = P (0, 1/n). Now set

p,n := {g | g (Kp.n) Kp,n 6= } .

Then p,n+1 p,n, and moreover p,n is a finite set for each p X and n N, andnN

p,n = {idX}.

It follows that there exists n(p) such that for n n(p), p,n = {idX}. Choose an open set Upcontaining p and contained in Kp,n(p). Then g(Up) Up 6= implies g = idX .

This allows us to deduce that the quotient space X/ is Hausdorff. Indeed, given p, q X suchthat (p) 6= (q), we can find open disjoint neighborhoods Up, Uq of p and q respectively such thatany non-identity element of satsifies g(Up) Up = g(Uq) Uq = . It follows (Up) (Uq) = .Since pi is an open map, (Up) and (Uq) are open neighborhoods of (p) and (q) respectively,X/ is Hausdorff as claimed.

Now observe that pi|Up : Up (U[) is a homeomorphism. Shrinking U if necessary, we mayassume U is the domain of a chart hp : Up Op Cn. Now let kp := hp

(pi|Up

)1 : (Up) Op.Then kp is a chart on X/ about (p). This shows that X/ is a topological manifold of dimension2n.

Finally we claim that {((Up), kp)} is a holomorphic atlas for X/. Suppose (Up)(Uq) 6= .Then

kp k1q = hq (pi|Uq

)1 (pi|Up) h1p .The key thing to evaluate is

(pi|Uq

)1 (pi|Up). Given z Up Uq, if (pi|Uq)1 (pi|Up) (z) = wthen w = gz(z) for some gz , where gz could conceivably depend on z. But since the sets g(Up)do not overlap as g varies through , it follows that gz maps Up into one connected component of(Up), and hence gz g for z Up Uq, which verifies the claim. I


1.17 CorollaryThe Hopf manifold Hn introduced in Example 4 of Section 1.15 is indeed an n-dimensional complexmanifold which is homeomorphic to S2n1 S1.

J We will apply the previous Proposition. Take take X = Cn\{0}, and let = g = Z, whereg : X X is defined by g (z) := 2nz. Then the Hopf manifold is precisely X/. Moreover,given any neighborhood U of 0 in Cn, and any compact set K X, there exists n0 such thatgn0(K) U X. Since if K is any compact set of X we can find a neighborhood U of 0 in Cnsuch that K U = , the hypotheses of Proposition 1.16 are satisfied, and we conclude the Hopfmanifold is a complex manifold.

To establish a homeomorphism between X/ and S2n1 S1, map

(z) 7(z

|z| , e2pii log2|z|

) S2n1 S1,

where we view S2n1 as the unit sphere in Cn. This is well defined, as if w (z) then w/|w| = z/|z|and e2pii log2|z| = e2pii log2|w|. This is clearly continuous, and has inverse the map (z, ei) 7 (2z),which is well defined for the same reason. I

1.18 LemmaEvery complex manifold is canonically orientated by its holomorphic atlas.

J Immediate from Lemma 1.9, as det JR(h h1

)> 0 for each overlapping set of charts. I

1.19 The complexified tangent spaceLet Xn be a complex manifold. Given p X, let (z1, . . . , zn) be local coordinates about x, andwrite zi = xi + iyi. By definition the real tangent space at p is simply the tangent space at pto the real manifold XR, Tp (XR). This a real 2n-dimensional vector space with basis{

x1p, . . . ,

xmp,

y1p, . . . ,

ymp

}We then consider the complexified tangent space

TC,p (X) := Tp (XR)R C.That is,

Tp (XR) = R{

xip,

yip

}, TC,p (X) = C

{

xip,

yip

}.

We letTC(X) :=

pX

TC,p (X)

denote the complexified tangent bundle. The same proof as in the real case shows that TC(X)is a complex vector bundle of rank n over X.

If we define a map Jp : Tp (XR) Tp (XR) by

Jp

(

xip

)=

yip, Jp

(

yip

)=

xip, (3)

then Jp End (Tp (XR)) and J2p = id. Now Jp extends complex linearly to give a map (alsodenoted by) Jp : TC,p (X) TC,p (X), and its eigenvalues are {i}. We let

T 1,0p (X) TC,p (X) := {v TC,p (X) | Jp(v) = iv} ,T 0,1p (X) TC,p (X) := {v TC,p (X) | Jp(v) = iv} .

For reasons that will shortly become clear, we call T 1,0p (X) the holomorphic tangent space atp and T 0,1p (X) the antiholomorphic tangent space at p. We define complex conjugation

TC,p (X) TC,p (X) to be the map sending v 7 v . Note that T 1,0p (X) = T 0,1p (X).


1.20 PropositionLet Xn be a complex manifold and p X. The dimC T 1,0p (X) = dimC T 0,1p (X) = n, and

TC,p (X) = T 1,0p (X) T 0,1p (X) . (4)Moreover the decomposition (4) is coordinate independent, and so is the map Jp defined by (3).Thus

T 1,0(X) :=pX

T 1,0p (X) and T0,1(X) :=

pX

T 0,1p (X)

are well defined n-dimensional complex subbundles of TC(X), and J defines well defined smoothglobal sections of the bundles End(T (X)) and End (TC (X)).

J Select local coordinates (z1, . . . , zn) about p, and observe that{

z1p, . . . ,

znp,

z1p, . . . ,

znp

}(given by (1)) forms a basis of TC,p(X) and

Jp

(

zip

)= i

zip, Jp

(

zip

)= i

zip. (5)

It thus follows easily that

T 1,0p (X) = C{

zip

}, T 0,1p (X) = C

{

zip

}and dimC T 1,0p (X) = dimC T 0,1p (X) = n, and TC,p (X) = T 1,0p (X) T 0,1p (X) . Moreover, since afunction f : U C (where U is some neighborhood of p) is holomorphic if and only if fzi 0,we can realise T 0,1p (X) as the subspace of TC,p(X) consisting of deriviations of Cp that vanish onOp Cp (where Cp denotes the ring of germs of smooth complex valued functions at p), and thislatter description is manifestly coordinate independent. Hence T 0,1p (X), and thus also T 1,0p (X) areindependent of the choice of holomorphic coordinates.

Finally, we have shown that the eigenspaces of Jp are intrinsically defined, and hence so is Jp.To check that the map J : T (XR) T (XR) defined by J |Tp(XR) := Jp is a well defined section ofEnd (T (X)), it remains to show p 7 Jp is smooth. But since

J

(ai

xip+bi

yip

)= bi

xip+ai

yip,

in a local trivialisation of T (XR), J is given by(p, a1, . . . , an, b1, . . . , bn

) 7 (p,b1, . . . ,bn, a1, . . . , an) ,which is visibly smooth. I

Note that we can also define

T 1,0p (X) := {v iJ(v) | v Tp (XR)} ,T 0,1p (X) := {v + iJ(v) | v Tp (XR)} ,

and thus we have an isomorphism

: T (XR) T 1,0(X), (v) = v iJ(v), (6)which will prove to be useful later. Note that

(Jv) = i(v), (7)

that is, intertwines J and i.A priori, T 1,0(X) is just an rank n complex vector bundle. We shall see in Chapter 4 that

T 1,0(X) actually admits the structure of a holomorphic vector bundle.


1.21 LemmaIf (z1, . . . , zn) and (w1, . . . , wn) are any two holomorphic coordinate systems on a complex manifoldX at p then

wjp=

zi

wj(p)

zip,

wjp=

zi

wj(p)

zip.

J Since{

zi

p, zi

p

}forms a basis of TC,p(X) we can write

wjp=

zi

wj(p)

zip+zi

wj(p)

zip,

and the second term is zero as the zi and the wj are holomorphic. Similarly wjp= z

i

wj (p)zi

p. I

1.22 Holomorphic vector fieldsWe let XR, XC, X 1,0 and X 0,1 denote the sheaves of sections of the vector bundles T (XR), TC(X),T 1,0(X) and T 0,1(X) respectively. Thus if U X is a coordinate domain with local holomorphiccoordinates (z1, . . . , zn) and zi = xi + iyi then:

V XR(U) V (p) = V i(p) xi

p+W i

yip, V i,W i C(U,R),

Z XC(U) Z(p) = Zi(p) zip+W i(p)

zip, Zi,W i C(U,C),

Z X 1,0(U) Z(p) = Zi(p) zip, Zi C(U,C),

Z X 0,1(U) Z(p) = W i(p) zip, W i C(U,C).

We let X 1,0hol X 1,0 denote the subsheaf of holomorphic vector fields on X; a local section Zof T 1,0 (X) is called a holomorphic vector field if the functions Zi : U C are holomorphic,Zi O(U) C(U,C).

1.23 The real representation of GL(n,C)The real representation of GL(n,C) in GL(2n,R) is given by

A+ iB GL(n,C) 7(A BB A

) GL(2n;R).

We will without further comment identify GL(n,C) with its image in GL(2n,R). We can charac-terise this image as

GL(n,C) = {A GL(2n,R) | AJ0 = J0A} , (8)where

J0 :=(

0 InIn 0

) GL(2n;R). (9)

1.24 The differentialLet : X Y be a smooth map between complex manifolds X and Y . Then induces adifferential d : T (XR) T (YR), which can be extended complex linearly to give a map d :TC(X) TC(Y ).

2 Differential (p, q)-forms and Dolbeault cohomology 9

1.25 PropositionLet : X Y be a smooth map between complex manifolds X and Y . Then the following areequivalent:

1. is holomorphic,

2. d intertwines JX and JY , that is, JY d = d JX ,3. d

(T 1,0 (X)

) T 1,0 (Y ),4. d

(T 0,1 (X)

) T 0,1 (Y ).J The assertion is local, and so we reduce to the case X = U Cn, Y = V Cm with U, Vopen. Take coordinates zi = xi + iyi on U and wi = ri + isi on V . Then by (3) both JX and JYare given by the matrix J0 from (9) (although JX is an n n matric and JY an m m matrix).d is given by the matrix JR, and examining (2) it is clear that JR takes values in GL(n,C) ifand only if is holomorphic. Thus by (8) we have the equivalence of the first two statements.

The second statement implies the third and fourth ones, since JR commuting with J0 impliesthat JR preserves the eigenspaces of J0, and hence d preserves the splitting of TC(X). Complexconjugation implies the equivalence of the last two statements, and both of these together implythe second statement. Indeed, since JX |T 1,0(X) = iid, JY |T 1,0(Y ) = iid, and similarly JX |T 0,1(X) =iid, JY |T 0,1(Y ) = iid, as everything commutes with iid, it follows that if d preserves thesplitting then d interwines JX and JY when restricted to either T 1,0 (X) or T 0,1 (X), and thenas TC (X) = T 1,0 (X) T 0,1 (X) the statement follows. I

2 Differential (p, q)-forms and Dolbeault cohomology

2.1 Differential forms of type (p, q)Let X be a complex manifold with underlying real manifold XR.We can also complexify the cotan-gent bundle T (XR) to form T C(X) := T

(XR) R C. The map J : T (XR) T (XR) induces adual map (which by an abuse of notation we also denote J), J : T (XR) T (XR), and hencea complex linear extension J : T C(X) T C(X), and we define T (X)1,0, T (X))0,1 to be the ieigenspaces of J . We call T (X)1,0 the holomorphic cotangent bundle. We can also form theexterior power bundles r (T C(X)), and if we set

p,q (X) := p(T (X)1,0

) q (T (X)0,1)then we have the decomposition

r (T C(X)) =p+q=r

p,q(X).

Note thatp,q(X) = q,p(X).

We let rC denote the sheaf of sections of r (T C(X)); thus given U X, rC(U) is the set of

complex differential r-forms. Similarly we let p,q denote the sheaf of sections of p,q(X);these are the differential (p, q)-forms. If U X is a coordinate neighborhood, with holomorphiclocal coordinates (z1, . . . , zn) then p,q(U) implies

=

i1


When p = q, there is a non-trivial real subspace

p,p(X) 2p (T (XR))

consisting of (p, p)-forms invariant under complex conjugation; these are the real (p, p)-forms, andwe let R denote this sheaf. Finally we let

phol p,0 denote the sheaf of holomorphic p-forms,

whose coordinate functions are holomorphic; that is, if U X is a coordinate neighborhood, withholomorphic local coordinates (z1, . . . , zn) then phol(U) implies

=

i1


2.4 The conjugate differentialSimilarly we define dc = i( ). Then = 12 (d+ idc) and = 12 (d idc). dc enjoys the sameproperty that d does (and and dont); dc is real operator, that is, dc (rR(X)) r+1R (X).Using the previous lemma, we have (dc)2 = 0, and ddc = dcd = 2i.

Note that if : X Y is a smooth map between complex manifolds then the pullback : r (YR) r (XR) can be complex linearly extended to define a pullback map : rC(Y )rC(X).

2.5 PropositionIf : X Y is a smooth map between complex manifolds then : rC(Y ) rC(Y ) preservesthe type (p, q) if and only if is holomorphic.

J Suppose that is holomorphic. Let 1,0(Y ), and let Z XC(X). Then since JY () =i by (10), we compute:

JX () (Z) = (JX(Z))= (d (JX(Z))()= (JY (d(Z))= JY ()(d(Z))= i(d(Z)) = i(Z),

where in () we used Proposition 1.25. Thus 1,0(X). In exactly the same way we see thatif 0,1(Y ) then 0,1(X), and then since any p,q(Y ) is a sum of wedge products offorms in 1,0(Y ) and 0,1(Y ), and preserves the wedge product, the result follows. Converselyif preserves type then since the above computation works for every complex vector field Z, ()shows that d intertwines JX and JY , and then Proposition 1.25 tells us that is holomorphic. I

We know that commutes with d in the real case; thus the complex linear extensions alsocommute. In the holomorphic case we have shown more:

2.6 CorollaryLet : X Y be a smooth map between complex manifolds. Then commutes with if andonly if is holomorphic.

J Proposition 2.5 is precisely the statement that commutes with the projection operatorsp,q if and only if is holomorphic, and clearly this is equivalent to commuting with :

Y () = p,q+1 dY () = p,q+1 dX = X () . IOf course, a similar statement holds for , but this is less important, as the following shows.

2.7 DefinitionThe Dolbeault complex is the cochain complex

. . . p,q1(X) p,q(X) p,q+1(X) . . .

which does indeed form a cochain complex thanks to Lemma 2.3.2. The associated cohomologyHp, (X)is called the Dolbeault cohomology of X.

Corollary 5.6 implies that any holomorphic map : X Y induces a contravariant functor : Hp,(Y ) Hp,(X). In particular, if X and Y are biholomorphic then Hp,(X) = Hp,(Y ).

We can also form the complex de Rham complex {rC, d}. As in the real case, the de Rhamcohomology is a topological invariant of X. However the Dolbeault cohomology Hp,(X) depends


on the holomorphic atlas of X and so is an invariant of the complex manifold. Despite the factthat rC(X) =

p+q=r

p,q(X), in general,p+q=r

Hp,q(X) 6= HrC,dR(X),

whereHC,dR(X) := H

dR (XR)R C,

but if M is compact and Khler then this does hold; this is called the Hodge decompositiontheorem and will proved in Chapter 10 - see Theorem 10.9

We now wish to prove the -Poincar Lemma, which is a version of the Poincar Lemma forDolbeault cohomology. Before doing so, we need a generalisation of the Cauchy integral formula.

2.8 Proposition (the extended Cauchy integral formula)Let D = D(a, r) be a disc in C, and let f C (D), and let z D. Then

f(z) =1

2pii

D

f(w)w z dw +

12pii

D

f

w(w)

dw dww z .

Note that if f is holomorphic then fw = 0, and so the second term vanishes and we recover thestandard Cauchy integral formula.

J Let D := D(z, ) and = 12piif(w)wz dw 1C (D\D). Then

d = =12pii

f

w

dw dww z ,

and hence by Stokes Theorem we have

12pii

D

f(w)w z dw =

12pii

D

f(w)w z dw +

12pii

D\D

f

w

dw dww z .

First, we claim that1

2pii

D

f(w)w z f(z) as 0.

To see this, change variables by setting w z = rei to observe that

12pii

D

f(w)w z =

12pi

2pi0

f(z + ei)d.

Since f is smooth, the integral on the right tends to f(z) as 0.Secondly, since dw dw = 2idx dy = 2irdr d, we have fw (w)dw dww z

= 2 fw dr d C |dr d| ,

and hence fwdwdwwz is absolutely integrable over D, and hence we conclude

12pii

f

w

dw dww z 0 as 0.

The result follows. I


2.9 Theorem (the -Poincar Lemma in one complex variable)Let D = D(a, r) be a disc in C, and let g C (D). Then the function

f(z) :=1

2pii

D

g(w)w z dw dw

is defined and is in C(D), and satisfies

f

z= g.

J Pick z0 D, and choose > 0 such that D2 := D(z0, 2) D. Using a partition of unityfor the open cover {D\D, D2} of D write

g(z) = g1(z) + g2(z),

where g1 vanishes outside of D2 and g2 vanishes on D, and then define

f2(z) :=1

2pii

D

g2(w)w z dw dw.

Then f2 is certainly defined and smooth on D, since the numerator vanishes when w approachesz then. Since f2 is smooth, we may differentiate under the integral to obtain

f2z

(z) =1

2pii

D

z

(g2(w)w z

)dw dw

for z D. But

z

(g2(w)w z

)= 0,

since g2(w) is independent of z and (w z)1 is holomorphic on C\{w}.Next, since g1(z) has compact support, we can write

12pii

D

g1(w)dw dww z =

12pii

Cg1(w)

dw dww z

=1

2pii

Cg1(u+ z)

du duu

= 1pi

Cg1(z + rei

)eidr d =: f1(z),

which is clearly defined and smooth in z. Hence again we may differentiate under the integral signand discover

f1z

(z) = 1pi

C

g1z

(z + rei

)eidr d

=1

2pii

g1w

(w)dw dww z .

But now we apply the extended Cauchy integral formula (Proposition 2.8) to see that

g1(z) =1

2pii

D

g1(w)w z dw +

12pii

D

g1w

(w)dw dww z =

f1z

(z),

since the first term vanishes as g1 is compactly supported in D2.Then we finally have f = f1 + f2 smooth on D and for z D

g(z) = g1(z) =f1z

(z) =f

z(z).

Finally, since z0 was arbitrary the stated result follows. I


2.10 Theorem (the -Poincar Lemma - general version)Let P = P (a, r) Cn denote a polydisc, where any of the ri may in fact be infinite (so P couldbe all of Cn). Then for all q 1, we have

Hp,q(P ) = 0.

J We first reduce to the case p = 0. Indeed, if p,q(P ) is closed then we may write

=|I|=p

I dzI ,

where I 0,q(P ). Then if we can find I such that I = I then

|I|=p

I dzI =

|I|=pI dzI = .

Hence assume 0,q(P ). The proof will break into two main steps.Step 1: The first step is to show that if P = P (a, s) where si < ri for all i then we can find

0,q1(P ) such that = |P . To see this write

=|I|=q

I dzI .

We say 0 modulo dz1, . . . , dzk if I 0 unless I {1, . . . , k}. We will prove the followingclaim k: if 0 modulo dz1, . . . , dzk then there exists 0,q1(P ) such that 0modulo dz1, . . . , dzk1. Since the hypotheses of the base case k = n are clearly satisfied, repeatingthis n times then completes the proof of the first step.

So suppose 0 modulo dz1, . . . , dzk and write

= 1 dzk + 2,

where both 1 and 2 contain no terms with dzk, that is,

1 =

|I|=q, kII dzI\{k},

2 =

|I|=q, k/II dzI ,

and so 2 0 modulo dz1, . . . , dzk1. Note that if ` > k then 2 contains no terms with factordz` dz`, and since = 1 + 2, it follows that

Iz`

= 0

for ` > k and I such that k I. Now set

=

I : kIIdz

I\{k},

whereI(z) :=

12pii

|w|sk

I(z1, . . . , zk1, w, zk+1, . . . , zn

) dw dww zk .

Then by the -Poincar Lemma in one complex variable (Theorem 2.9) we have

Izk

(z) = I(z),

3 Almost complex manifolds 15

and for ` > k,

Iz`

(z) =1

2pii

|w|sk

Iz`

(z1, . . . , zk1, w, zk+1, . . . , zn

) dw dww zk = 0.

Hence 0 modulo dz1, . . . , dzk1 as claimed.Step 2: The second step is to improve the result to obtain a primitve for on all of P . Let(

rkj)be a monotone increasing sequence such that rkj rk as j for all k = 1, . . . , n, and let

Pj = P (a, rj). By the above, we can find j 0,q (Pj) with j = |Pj . We need however to dobetter than this; we wish to modify the (j) to get new forms (j), where j = j + j so thatthey converge in an appropiate sense to some 0,q(P ) such that = , which will completethe proof.

We will induct on q, and leave the base case till last, so suppose initially q 2, and that wehave chosen i. Then observe that (i i+1) = 0 on i. By induction on q, we may assumethus assume that ii+1 = i+1 on Pi1. Now extend i+1,i and i+1 smoothly in some waysuch that they are defined on all of P , and set i+1 = i+1 + i+1. Then i+1 = i+1 = onPi+1, and moreover on Pi1 we have i+1 = i. Thus the modified sequence (j) stabilise on allcompacta, and hence we obtain our desired limit form .

It remains to consider the case where is a (0, 1)-form. In this all the (j) are functions.Suppose we have chosen i. Extend i and i+1 smoothly so that they are both in C(P ). Thenobserve (i i+1) = 0 in Pi, and hence ii+1 is a holomorphic function in Pi. Hence it hasa power series expansion about a which is valid on Pi and converges uniformly with all derivativeson Pi1. Truncating this sum we obtain a polynomial i+1 such that

supzPi1

|(i i+1) i+1| < 2i.

Extend i+1 smoothly to Pi+1, and set i+1 := i+1 + i+1 (not i+1 + i+1; this is different tothe previous case!). Then i+1 = i+1 = on Pi+1,and the function i+1 i is holomorphicin Pi, with

supzPi1

|i+1 i| < 2i.

It follows that the function := lim j exists and = on P . This completes the proof. I

3 Almost complex manifolds

3.1 DefinitionLet M be a smooth real manifold. An almost complex structure J on M is a section of thebundle End(T (M)) such that J2 = id. We call the pair (M,J) an almost complex manifold.

Proposition 1.20 shows that if X is a complex manifold then (XR, J)is an almost complexmanifold, where J is given by (3).

3.2 LemmaAny almost complex manifold is even dimensional.

J Suppose V is an m-dimensional real vector space admitting an endomorphism J : V Vsuch that J2 = id (in general such an endomorphism is called a complex structure; thus analmost complex manifold is a smooth manifold admitting a smoothly varying complex structureon its tangent spaces). We prove m is even by induction. We may assume m > 0. Select v1 6= 0.Then {v1, J (v1)} is linearly independent. Indeed, if

v1 + J(v1) = 0,

then applying J to both sides and writing the equations in matrix form we obtain(

)(v1

J (v1)

)=(

00

).


But the matrix has determinant 2 +2, and thus we obtain = = 0. Note that span{v1, J(v1)}is J-invariant. If this spans V we are done. Otherwise observe that the complement V :=V \span{v1, J(v1)} is also J-invariant, and hence J restricts to define a map J |V : V V suchthat (J |V )2 = id. Thus by induction on the dimension, V admits a basis of the form

{v1, . . . , vn, J (v1) , . . . , J (vn)} , 2n = m.

Thus V is even dimensional as claimed; this clearly implies the lemma. I

3.3 Parametrizing the complex structures on R2n.We can parametrize the set J of complex structures on R2n by the set of cosetsGL(2n,R)/GL(n,C)via [A] ! J[A] := AJ0A1. This is well defined since if [A] = [B] then A = BC for someC GL(n,C) and then AJ0A1 = BCJ0C1B = BJ0B1, since C commutes with J0. Note alsothat J[A] is clearly a complex structure. Moreover, for any complex structure J on R2n there exists[A] such that J = J[A]; indeed simply select a basis as in Lemma 3.2, and arrange the basis in theorder {v1, . . . , vnn, J(v1), . . . , J(vn)}. Then the change of basis matrix A is the desired matrix.

We have thus proved:

3.4 LemmaIf (M,J) is any almost complex manifold and p M , there exists a local frame about p such thatJ is represented by the matrix J0.

3.5 PropositionEvery almost complex manifold is orientable.

J If (M,J) is an almost complex manifold by Lemma 3.2 (applying to V = T p (M) and J theinduced dual map T (M) T (M)) we may choose an open cover {U} of M such that thereexists a frame {i , J (i )} of T (U). Then if U U 6= ,

1 mJ (1 ) J (m) = det {change of basis matrix}1 mJ(1

) J (m) ,

and since the change of basis matrix is non-singular and lies in GL(m,C), it has positive determi-nant. Thus M admits a volume form (given locally as 1 m J (1 ) J (m)) andthus is orientable. I

3.6 DefinitionLet (M,J) be an almost complex manifold. TheNijenhuis tensor of the almost complex structureJ is NJ

(Hom(2(T (M)), T (M)

)defined by

NJ(X,Y ) := [JX, JY ] [X,Y ] J ([JX, Y ]) J ([X.JY ])

for X,Y local vector fields. We say that J is torsion free if NJ 0. To check NJ does indeeddefine a tensor:

3.7 Lemma

NJ is linear over C(M), and locally NJ depends only on the coordinates J ij andJijxk

.We use the fact that [fX, Y ] = f [X,Y ] Y (f)X.


Take f (M). Then

NJ(fX, Y ) = [fJX, JY ] [fX, Y ] J ([fJX, Y ]) J ([fX, JY ])= f [JX, JY ] JY (f)JX f [X,Y ] + Y (f)XJ (f [JX, Y ] Y (f)JX + f [X, JY ] JY (f)X)

= f ([JX, JY ] [X,Y ] J ([JX, Y ]) J ([X.JY ]))JY (f)JX + Y (f)X + JY (f)JX + J2Y (f)X

= fNJ(X,Y ).

For the second assertion, let (x1, . . . , x2m) be local coordinates on M and choose X = i andY = j . We obtain

NJ (i, j) =: Nkij

xk,

where

Nkij =

(J`iJkjx` J`j

Jkix` Jk`

J`jxi

+ Jk`J`ixj

). I (12)

3.8 DefinitionWe say an almost complex structure J on a manifold M2n is integrable if it arise from a holo-morphic atlas on M ; that is, M = XR admits the structure of an n-dimensional complex manifoldX, and J is defined by (3).

The following theorem is cruical. The proof of the full theorem is extremely hard - we will proveone direction only. We note however that the other direction is much easier to prove under theassumption M is analytic, where then the theorem follows readily from the Frobenius integrabilitytheorem.

3.9 Theorem (Newlander-Nirenberg)An almost complex structure J is integrable if and only if it is torsion free; NJ 0.newlineJ (we prove only) Suppose J arises from a holomorphic atlas. If (z1, . . . , zn) are

coordinates on U then writing zi = xi + ixn+i then

J (i) =

{i+n 1 i nin n+ 1 i 2n,

and thus from (12) it is clear NJ 0. I

Note that even if J is not integrable, we can still decompose TC (M) := T (M) R C intoT 1,0 (M) T 0,1 (M), since the decomposition only relied on the existence of the map J , not theholomorphic atlas. Similarly we can decompose r (M) into

p+q=r

p,q (M) and we can stillobtain the operators and . However we do lose something by not assuming integrability, asProposition 3.11 will make precise.

3.10 PropositionLet (M,J) be an almost complex manifold. Then

d (p,q(M)) p1,q+2(M) p,q+1(M) p+1,q(M) p+2,q1(M).

The result clearly holds for p = q = 0. Similarly, since 2C(M) = 0,2(M)1,1(M)2,0(M),

the result certainly holds for 1,0(M) and 0,1(M). Since the assertion is local and p,q is generatedlocally by 0,0,1,0 and 0,1, the result follows. I


3.11 Proposition (characterisations of integrability)Let (M,J) be an almost complex manifold. Then the following are equivalent:

1. For all local sections Z,W of T 1,0 (M), [Z,W ] is also a local section of T 1,0 (M).

2. For all local sections Z,W of T 0,1 (M), [Z,W ] is also a local section of T 0,1 (M).

3. d(1,0(M)

) 0,0(M) 1,1(M).4. d

(0,1(M)

) 0,0(M) 1,1(M).5. d (p,q(M)) p,q+1(M) p+1,q(M).6. d = + .

7. NJ 0.8. J is integrable.

J First the easy bits. The first two statements are clearly equivalent by complex conjugation.Similarly so are the third and fourth statements. Statements (3) and (4) imply (5) since p,q islocally generated by 0,0,1,0 and 0,1, and clearly (5) implies (3) and (4). The fifth statement isequivalent to the sixth by definition. The equivalence of the last two statements is the Newlander-Nirenberg theorem.

To complete the proof we show that the first statement is equivalent to both the fourth andthe seventh. If is an arbitrary complex-valued 1-form and Z and W are arbitrary complexvector fields of type (0, 1) then since

d(Z,W ) = Z(W )W(Z) ([Z,W ]) ,we obtain

d(Z,W ) = ([Z,W ]) . (13)Then the first statement is precisely that the right-hand side of (13) is always zero, and the fourthstatement is precisely that the left-hand side of (13) is always zero.

Finally, let X and Y denote arbitary real local vector fields, and set

Z := [X iJX, Y iJY ].Direct computation shows

Z + iJZ = NJ(X,Y ) iJ (NJ(X,Y )) . (14)Recalling the bijection (6), the first statement is precisely that the left-hand side of (14) is

always zero, and the seventh statement is precisely that the right-hand side of (14) is always zero.This completes the proof. I

The existence of an almost complex structure on M2n is a topological question, whereas theintegrability of an almost complex structure J on M is a question in geometric analyis, namely, anonlinear 1st order PDE on a certain vector bundle. It can be shown that Sn admits an almostcomplex structure if and only if n = 2 or 6, and it is not known whether S6 admits an integrablealmost complex structure. We conclude this discussion with the following result.

3.12 PropositionEvery orientated real surface S admits an almost complex structure, and all such almost complexstructures are integrable.

J Let g be a Riemannian metric on S. Let p S. Then we can find a neighborhood U of pand isothermal coordinates (x1, x2) on U such that gij(q) = (q)ij for all q U . Interchangingx1 and x2 if necessary, we may assume {1, 2} is a positive frame on U .

4 Holomorphic vector bundles, subvarieties and some commutative algebra 19

We define JU : T (U) T (U) by

JU (1) = 2, JU (2) = 1.

Thus JU is given by the matrix J0 with respect to the basis {1, 2} of T (U). We want to usethis definition to define J on all of T (S). Suppose (y1, y2) are positively oriented isothermal localcoordinates on another neighborhood V of p. The change of basis matrix B then lies in SO(2);since SO(2) GL(1,C) GL(2,R), B commutes with J0, whence it follows that JU = JV on theoverlaps, and so this defines an almost complex structure on S.

To see that J is necessarily integrable, simply observe that any local vector field X X (U)can be written as a linear combination of 1 and 2, and since one easily checks

NJ (1, 1) = NJ (1, 2) = NJ (2, 2) = 0,

we have NJ 0 and thus by the Newlander-Nirenberg Theorem 3.9 it follows J is integrable. I

4 Holomorphic vector bundles, subvarieties and some commutative algebra

4.1 DefinitionLet X be a complex manifold. A holomorphic vector bundle of rank m over X is a complexmanifold E with a holomorphic submersion pi : E X (i.e. dpi(p) is surjective for all p X)such that for all p X, Ep := pi1(p) admits the structure of a complex vector space, and thatthere exists a neighborhood U of x and a biholomorphic map t : pi1(U) U Cm such thatpi = pr1 t : pi1(U) U and such that tp := t|Ep : Ex Cm is a complex linear isomorphism.

If U and U are two overlapping holomorphic trivialising neighborhoods of X for E and tand t the corresponding holomorphic local trivialisations than

t t1 : (U U) Cm (U U) Cm

is a map of the form(p, v) 7 (p, (p)(v)) ,

where : U U GL(m,C) is a holomorphic transition function. The {} satisfythe cocycle conditions = id, = 1 and = id.

A holomorphic local section of E over U X is a holomorphic map s : U E such thatpi s = id. We let OE denote the sheaf of holomorphic sections of E, and EE the smooth sections,so OE EE .

4.2 Examples1. Let Xn be a complex manifold. Then T 1,0(X) X is a holomorphic vector bundle of rank n.

Indeed, if {U, h} is a holomorphic atlas forX then T 1,0(X) has cocycle{U, J

(h h1

) h

};

since p 7 J(h h1

)h(p) is holomorphic this gives T 1,0(X) the structure of a holomor-

phic vector bundle.

2. The canonical bundle KX = n,0(X) is a holomorphic line bundle.For this, we observe T (X)1,0 is a holomorphic vector bundle (being the dual of T 1,0(X))and thus KX = detT (X) is also holomorphic. Explicitly KX has holomorphic cocycle{U, } where

={

det J(h h1

) h

}1.

4.3 PropositionThe isomorphism classes of holomorphic line bundles over a complex manifold X form an abeliangroup under called the Picard group of X, denoted Pic(X).


J We define the product of two line bundles L,L on X to be the tensor product LL. Thetrivial line bundle is the identity element. The only thing that needs a proof is that L L isisomorphic to the trivial bundle, whence then the inverse to a bundle L is its dual bundle. This isbest seen by observing that if L is given by the cocyle {U, } then L is given by the cocycle{U, (t)1}, and thus L L is given by the cocycle {U, (t)1}. But then since L isof rank 1, t = , and we conclude that L L has trivial transition functions, and hence isisomorphic to the trivial line bundle. I

4.4 CorollaryIf : X Y is a holomorphic map then induces a group homomorphism : Pic(Y ) Pic(X)given by (L) = L.This is a contravariant functor.

J The assignment is clearly functorial. To check that is a group homomorphism, ob-serve that if L has cocycle {U, } and L has cocycle {U, } then (L L) has cocycle{U,

(

) }and LL has transition functions

{U ( )

(

)}, which

are visibly the same. I

4.5 DefinitionThe set O(1) CPn Cn+1 defined by

O(1) := {(z, w) Cn+1 CPn | z w}is called the tautological line bundle over CPn.

4.6 LemmaO(1) forms in a natural way a holomorphic line bundle over CPn.

J We define pi : O(1) CPn to be projection onto the second factor. Let CPn = ni=0 Uibe the standard open covering of CPn. A canonical trivialisation of O(1) over Ui is given by

ti : pi1(Ui) Ui C, (z, w) 7(w, zi

).

The transition functions ij(w) : C C are given by z 7 wi/wjz, where w = (w0 : : wn).This gives us a well defined line bundle; to complete the proof we check that O(1) is a complexmanifold of dimension n+ 1 in its own right. For this let hi : Ui = Cn denote the standard chart

hi :(w0 : : wn) (w0

wi, . . . , i, . . . ,

wn

wi

).

Then we can define charts hi : pi1 (Ui) C Cn by to be the composition (hi id) ti. I

4.7 DefinitionWe let O(1) := O(1) denote the hyperplane bundle that is the dual of the tautological bundle(see Proposition 6.20 for an explanation of the name hyperplane bundle). We let O(k) = O(1)kand O(k) = O(1)k for k > 0. If we set O(0) to be the trivial bundle CPn C then we havean injective group homomorphism Z Pic(CPn) given by k 7 O(k). We shall see later that thisin fact surjective, and hence deduce that Pic(CPn) = Z (see Corollary 6.19).

4.8 DefinitionLet Xn be a complex manifold and Y X a smooth embedded submanifold of real dimension2k (k n). Then Y is a complex submanifold if there exists a holomorphic atlas {(U, h)}of X such that h : U Y = h(U) Ck, where we consider Ck Cn via (z1, . . . , zk) 7(z1, . . . , zk, 0, . . . , 0). The codimension of Y in X is dimCX dimC Y = n k.


Equivalently (use the implicit function theorem), a subset Y X is a complex submanifold ofdimension k if for all p X there exists a neighborhood U of p in X and a holomorphic functionsf1, . . . , fk : U C such that 0 is a regular value for each fi and

Y U =ki=1

f1i (0)

(note that if p / Y this vacuously holds).We can rephrase Section 1.14 by saying that a complex manifold X is projective if it is biholo-

morphic to a closed complex submanifold of CPN for some N .

4.9 DefinitionsLetX be a complex manifold. An analytic subvariety ofX is a closed subset Y X such that forany p X there exists a neighborhood U of p in X and a holomorphic functions f1, . . . , fk : U C(where now k could depend on x) such that Y U = ki=1 f1i (0). The main difference is that weno longer require 0 to be a regular value of the fi.

If Y is an analytic subvariety of a complex manifold X then a point p Y is called a regularpoint if we can choose the functions fi : U C above such that 0 is a regular point on all ofthem. In other words, such that Y U is a complex submanifold of X. If p is not a regular pointit is called a singular point. We denote the set of all singular points of Y , called its singularlocus by Y s. If Y s = then Y is nonsingular or regular. We denote by Y the complementY \Y s. The implicit function theorem implies that every connected component of Y is a complexsubmanifold of X.

4.10 DefinitionAn analytic subvariety Y of a complex manifold M is called irreducible if it cannot be writtenas Y1 Y2 where Y1 and Y2 are two analytic subvarieties of M such that Y 6= Y1 and Y 6= Y2. IfY is irreducible, we define its codimension to be the codimension of the complex submanifoldY \Y s. More generally, we can define the codimension of an analytic subvariety if each connectedcomponent of Y \Y s has the same codimension. If Y has codimension 1 then Y is called a analytichypersurface; equivalently this is if and only if Y can locally be expressed as the zero locus of aholomorphic function f : U C.

Let On := OCn,0. We now six results that are proved by a mixture of complex analysis andcommutative algebra; for the purposes of this course we shall just assume them. They concern theproperties of the local ring On, and more generally, the local rings OX,p.

4.11 Theorem (Weierstrass preparation theorem)Write the coordinates of Cn as (w, z) CCn1 = Cn. Suppose f : Cn C is holomorphic near0, f(0, 0) = 0 and f(, 0) is not identically zero. Then there exist unique holomorphic functions g, hon a (possibly smaller) neighborhood of 0 such that on this neighborhood f = gh, with h(0, 0) 6= 0and g a Weierstrass polynomial. By this we mean

g(w, z) = wd + ad1(z)wd1 + + a1 (z)w + a0 (z) ,

where the ai are holomorphic on some neighborhood of 0 Cn1 and satisfy ai(0) = 0. The integerd is called the degree of g.

In the special case that n = 1, this reduces to the well known result from complex anaylsis thatif f : C C is holomorphic at 0 with f(0) 6= 0 then there exists a unique holomorphic functionsh with h(0) 6= 0 such that on a neighborhood of 0, f(w) = wdh(w) for some integer d, called theorder of f at 0 and denoted ordf (0).


4.12 CorollaryThe number of zeros of a holomorphic function is stable under holomorphic perturbation.

4.13 Theorem (Weierstrass division theorem)Let g On be a Weierstrass polynomial of degree d. Suppose f On. Then there exists uniqueh, r On with r a Weierstrass polynomial of degree strictly less than d such that on a suitablysmall neighborhood of 0, f = gh+ r.

4.14 Theorem (Weak Nullstellensatz)Let f, h On such that f is irreducible and such that there exists a neighborhood U of 0 on whichboth f and h are defined, and {f = 0} U {h = 0} U . Then f divides h in On, that is, h/f isholomorphic near 0.

4.15 Theorem (UFD)On is a unique factorization domain. Thus if Xn is a complex manifold and p X then Op = OCn,0is a unique factorization domain.

4.16 PropositionLet X be a complex manifold and p X.

1. Suppose f Op is a prime germ. Then there exists a small neighborhood V of p such thatf represents a prime germ in Oq for all q V .

2. If f and g are coprime elements of Op then there exists a small neighborhood V of p suchthat f and g are coprime in Oq for all q V .

4.17 DefinitionLet X be a complex manifold and Y X an analytic hypersurface. Then given p Y , there existsa neighborhood U of p in X and a holomorphic function f : U C such that Y U = f1(0). fis called a local defining function for Y at p. Suppose f = f1 . . . fn as a product of irreducibles.Then if g = g1 . . . gm is another local defining function for Y at p, then the Weak Nullstellensatzimplies that up to units, {f1, . . . , fn} = {g1, . . . , gm} and hence n = m and thus the local definingfunction at p is uniquely defined up to units. Moreover since each fi represents a prime germ inOX,q for q close to p, by Proposition 4.15.1, shrinking U if necessary we may assume that {fi = 0}is irreducible.

4.18 DefinitionLet Y X be an analytic hypersurface and p Y . We say that Y is locally irreducible at p ifthere exists a neighborhood U0 of p in X such that if U U0 is any other neighborhood of p in Xthen Y U is irreducible.

4.19 LemmaY is locally irreducible at p if and only if its locally defining function f at p is irreducible in OX,p.

J Suppose the local defining function f at p is reducible, say f = gh. Then for any neighbor-hood U of p in X small enough such that all f, g, h are defined, we have Y U = {g = 0}{h = 0}and thus Y is not locally irreducible at p.

Conversely suppose that f is irreducible in OX,p, but Y is not locally irreducible at p. Thenwe can find a neighborhood U of p in X such that Y U = Y1 Y2, where Y1 and Y2 are twoanalytic hypersurfaces, and p Y1 Y2. Let g and h be two local defining functions for Y1 and Y2


at p respectively. Then since for q U , g(q) = 0 implies f(q) = 0 and similarly h(q) = 0 impliesthat f(q) = 0, we have g|f and h|f in OX,p by the Weak Nullstellensatz, and hence either g or his a unit in OX,p. But then either g(p) 6= 0 or h(p) 6= 0, and this contradicts p Y1 Y2. I

4.20 LemmaLet X be a complex n-manifold and Y X an irreducible analytic hypersurface. Then if p Y isa regular point then Y is locally irreducible at p.

J The assertion that p is a regular point is equivalent to the existence of local coordinates(z1, . . . , zn) centred about p in X such that such that z1 is a local defining function for p at Y .Since z1 is clearly irreducible in On, the previous lemma completes the proof. I

We conclude this section by sketching the proof of an important result on the structure ofanalytic hypersurfaces.

4.21 TheoremLet Y be an analytic hypersurface of a complex manifold X. Then Y is a open dense set of Y .Moreover Y is connected if and only if Y is irreducible. Moreover Y s is contained in an analtyicsubvariety of codimension > 1.

J (sketch) If Y = Y1 Y2 with Y1, Y2 Y analytic subvarieties then Y1 Y2 Y s, so Y isdisconnected as Y \Y1 and Y \Y2 are open disjoint sets contained in Y . Now write Y =

i Yi

for the decomposition into connected components. One can show that Y is always non-empty anddense in Y , and each Zi = Yi is an irreducible analytic hypersuface of Y such that Zi * Zj if i 6= j.This implies that Y is the union of a irreducible analytic hypersurfaces, and the sum is locallyfinite in the sense that any p X is contained in a neighborhood U that intersects only finitelymany of the Zi. In particular, this shows that Y is irreducible if and only if Y is connected.

Now fix p Y , and let f be a local defining function for Y in a neighborhood U of p in X.Write f = fd11 . . . f

dkk where each fi is irreducible in OX,p. Replacing U by a smaller neighborhood

V of p in X, we have

V Y = {f = 0} =ki=1

{fi = 0}.

Each {fi = 0} V is a hypersurface in V . Since each fi represents a prime germ in OX,p, we havethat each fi defines a prime germ in OX,q for q near p by Proposition 4.16.2. Thus replacing V bya smaller neighborhood W of p in X we know that each {fi = 0} W is an irreducible analytichypersurface.

Observe that g := f1 . . . fk is also a local defining function for Y at p, and note that if h isanother local defining function for Y at p then g|h in OX,q for q near p. Thus dhq = 0 if dgq = 0for q, and hence

Y s W ={g =

g

z1= = g

zn= 0},

where (z1, . . . , zn) are local coordinates in W (perhaps after shrinking W again so that it is thedomain of a chart).

Suppose we could show that

{g = 0} *{g

z1= = g

zn= 0}. (15)

Then Y s is locally given by more than one holomorphic function, which implies that Y s is containedin an analytic subvariety of codimension > 1. But by the Weierstrass preparation theorem we mayassume that g is actually a Weierstrass polynomial of degree k in z1, say. Then since gz1 is then aWeierstrass polynomial of degree k 1 in z1, and it follows that g does not divide gz1 , and henceby the Weak Nullstellensatz we see that (15) holds. I

5 Meromorphic functions and divisors 24

5 Meromorphic functions and divisors

5.1 DefinitionLet X be a complex manifold and U X. A meromorphic function f on U is a map

f : U pUKp,

where Kp is the quotient field of Op such that for each p U there is an element fp Kp such thatfor any p U there exists a neighborhood V U of p and holomorphic functions g, h : V Csuch that for all q V , fq = g/h as elements of Kq.

Equivalently, a meromorphic function f on an open subset U X is specified by some opencovering {U} of U and holomorphic functions g, h : U C such that f |U = g/h such thatg and h are relatively prime elements in O (U) and gh = gh as elements of O (U U).We let K denote the sheaf of meromorphic functions on X, and K the multiplicative sheaf ofmeromorphic functions not identically zero.

A meromorphic function is not strictly a function since it is not defined (even if we consider a value) at points where both g = h = 0.

5.2 DefinitionLet X be a complex manifold, Y an irreducible analytic hypersurface and p Y a regular pointand f a local defining function for Y at p. Let g OX,p. Define the order of g along Y at p tobe the integer

ordY,p(g) := maxaN{0}

{a | fa divides g in OX,p}.

This is well defined, since a local defining function is unique up to units. It is finite, since OX,p isa unique factorisation domain. The following result is immediate from Proposition 4.16.1.

5.3 LemmaThe order of g is locally constant; there exists a neighborhood U of p in X such that if q U Ythen ordY,q(g) = ordY,q(g).

5.4 DefinitionIt follows that we can define the order of g along Y to be ordY (g) = ordY,p(g) for any regularpoint p Y , since Y is an open connected dense subset of Y by Theorem 4.20. The following isimmediate from the definitions: if g, h are holomorphic around p then

ordY (gh) = ordY (g) + ordY (h). (16)

5.5 DefinitionLet X be a complex manifold and f a meromorphic function on X that is not identically zero. LetY be an irreducible analytic hypersurface of X. We define the order of f along Y to be

ordY (f) = ordY (g) ordY (h),

where f = g/h at some regular point p Y . This is well defined by (16) and the compatibilitycondition on the representations f = g/h of f .

If d = ordY (f) > 0 then we say f has a zero of order d along Y , and if d < 0 then we say fhas a pole of order d along Y .


5.6 DefinitionsLet X be a complex manifold. A divisor D on X is a formal sum

D =i

aiYi,

where each Yi is an irreducible analytic hypersurface and each ai Z, such that D is locallyfinite; for any p X there exists a neighborhood U of p such that at most finitely many ai thatare non-zero and have U Yi 6= . We let Div(X) denote the abelian group of divisors on X. If Xis compact then any divisor D can be written as a finite sum of irreducible analytic hypersurfaces;thus if X is compact then Div(X) is simply the free Abelian group on the set of irreducible analytichypersurfaces. Say a divisor D =

i aiYi is effective if all the ai 0. This defines a partial

ordering on Div(X); namely D D if D D is effective.

5.7 DefinitionLet X be a complex manifold and f a meromorphic function on X not identically zero. We definethe divisor of f , written (f) to be

(f) =YX

irreducible hypersurface

ordY (f) Y.

This sum is locally finite (and hence is a well defined divisor), since given any p X there existsa neighborhood U of p such that there are only finitely many irreducible analytic hypersurfaces Ysuch that U Y 6= and ordY (f) 6= 0. Indeed, we may choose U such that on U , f = g/h with hnot identically zero on U ; the assertion is then clear, as g is not identically zero and hence both gand h vanish on at most finitely many hypersurfaces passing through U .

Note that (f) is effective if and only if f is a holomorphic function on X.

5.8 DefinitionWe say a divisor D that can written as (f) for some meromorphic function f is effective. Thisdefines a equivalence relation on Div(X), namely that D and D are linearly equivalent, writtenD D if D D is principal. Transitivity of this relation uses (16).

It is convenient to give a sheaf-theoretic definition of divisors. In what follows, we may regardsa non-vanishing holomorphic function as an invertible meromorphic function, and thus obtain aninclusion O K of sheaves. Thus we can consider the quotient sheaf K/O. It is convenientto use the notation H0 (X,K/O) for the global sections (X,K/O) of K/O.

5.9 PropositionLetX be a complex manifold. There is a natural isomorphism betweenH0 (X,K/O) and Div(X).

J An element k H0 (X,K/O) is given by an open cover {Ui} of X and non-trivialmeromorphic functions fi K(Ui) such that fi/fj O (Ui Uj) whenever Ui Uj 6= . IfY is an irreducible hypersurface with Y Ui Uj 6= we thus have ordY (fi) = ordY (fj) sinceordY (fi/fj) = 0 since fi/fj O (Ui Uj). Hence we may define

ordY (k) := ordY (fi)

for any Ui such that Y Ui 6= . This gives us a map

H0 (X,K/O) Div(X)k 7

Y

ordY (k) Y ;


the latter sum is well defined (locally finite) since the fi are non-trivial.By additivity of the order, this is clearly a group homomorphism. To show this is a bijection

we exhibit an inverse. Suppose D =i aiYi Div(X). Then there exists an open cover {U} of

X and functions gi O (U) such that

Yi U = g1i (0)

(if Yi U = then take gi to be any element of O (U)). Now set

f :=i

gaii K (U) ;

we may assume that this is a finite product. Since gi and gi define the same irreducible hyper-surface on U U , they differ only by an elements of O (U U). Thus the elements f gluetogether to define a global section k H0 (X,K/O). These two operations are clearly mutuallyinverse, and so this completes the proof. I

In what follows we will slightly abuse notation and refer to a divisor D has been given by localdata {U, f} to mean that D corresponds to k H0 (X,K/O) where k is specified by theopen cover {U} of X and f K(U).

5.10 Do divisors exist?If X is a compact Riemann surface then a divisor is just a collection of points, and hence X haslots of divisors. However if dimX > 1 there is no a priori reason for X to have any hypersurfaces,and hence any divisors. However if X is projective, say X CPN then if

H CPn :={(w0 : : wN) N

i=0

biwi = 0

}

then H is a hypersurface, and we can choose (bi) such that H X 6= , whence X H is ahypersurface of X. However there exist complex manifolds X such that Div(X) = but Pic(X) 6=; we shall shortly see the relevance of this.

5.11 Pullbacks of divisorsLet : Z X be a holomorphic map, with Z,X complex manifolds. Let D = i aiYi Div(X)be a divisor on X, and assume that (Z) Yi for i = 1, . . . ,m. We want to define a pullbackdivisor D Div(Z). The obvious definition D = i ai1(Yi) does not work, since 1(Yi)is not necessarily irreducible (among other problems).

However we can get around this problem by taking the sheaf-theoretic definition. Namely, if Dis given by local data {U, f} then we define D to be the divisor on Z given by local data{

1 (U) , f }.

It is easily seen that that D =mi=1 ai (

Yi). We call a holomorphic map : Z Xdominant if (Z) is dense in X. If is dominant then D is always well defined for anydivisor D Div(X), and thus a dominant map : Z X defines a group homomorphism : Div(X) Div(Z).

5.12 DefinitionsIf pi : E X is a holomorphic vector bundle, recall we let OE denote the sheaf of holomorphicsections of E. We now let KE denote the sheaf of meromorphic sections of E, where bydefinition

KE(U) := OE(U)OX(U) KX(U).


In particular if L is a line bundle with cocycle {U, } then a holomorphic section s OE(U)is given by holomorphic functions s O(U U) satisfying

s = s

on U U , and similarly a meromorphic section k KE(U) is given by meromorphic functionsk K(U U) satisfying k = k on U U .

Suppose k is a meromorphic section of L. Then if f is any meromorphic function on X thenclearly k = fk is another meromorphic section of L. Conversely if k and k are two meromorphicsections then letting f be the meromorphic function given by f = k/k then k = fk. Thus thespace of meromorphic sections of L is in bijective correspondence with the space of meromorphicfunctions on X (provided at least one meromorphic section exists).

Now note that if k is a non trivial meromorphic section then on U U , k and k have thesame zeros and poles (and the same orders) since is holomorphic and never zero. Hence if Yis an irreducible hyperspace of X, we can define the order of k along Y to be ordY U(k) forany such that U Y 6= . Thus we can define the divisor of a meromorphic section k, written(k) to be

(k) =YX

irreducible hypersurface

ordY (k) Y.

Then k is a holomorphic section of L if and only if (k) is effective.

5.13 TheoremThere exists a homomorphism [] : Div(X) Pic(X) whose kernel is precisely the principle divsors,and whose image is precisely the (isomorphism classes of) line bundles admitting a non-trivial globalmeromorphic section. Moreover [] is compatible with pullbacks in the sense that if : Z X isholomorphic, D Div(X) such that D is defined then [D] = [D].

J Let D be a divisor on X given by the data {U, f}. Let

:=ff,

and note that is holomorphic and non-zero on U U . It is immediate that {(U, )}satisfy the cocycle conditions and so defines an isomorphism class of holomorphic line bundles,that is, an element [D] of Pic(X). Let us check that the map D 7 [D] is well defined. For this it isenough to note that if {f } are alternative local data for D then f = sf for some holomorphicnever zero function s : U C. Thus the new transition functions are

= hh.

Now we claim that the holomorphic line bundle L given by the cocycle {U, s/s} is trivial. Indeed,it is enough to note that L admits a global non-vanishing holomorphic section s given locally bythe functions s.

Then if E denotes the holomorphic line bundle of D with respect to the old transition functionsand E the holomorphic line bundle of D with respect to the new transition functions then E =E L and hence the images in Pic(X) coincide. Thus [] is well defined.

It is clear that [] is a homomorphism, since if D1 and D2 are given by data {f1} and {f2}then D1 +D2 is given by the data {f1 f2}, whence it follows that [D1 +D2] = [D1] [D2].

Suppose D = (f), where f is a meromorphic function on X. Then we may take as local datafor D {U, f} where f is a quotient of holomorphic functions on U such that f |U = f. Butthen since f/f = id on U U , it follows [D] has trivial holomorphic cocycle and thus is thetrivial line bundle.

Conversely suppose that [D] is isomorphic to the trivial bundle. Then there exists a holomorphicnowhere vanishing section s. If [D] has cocycle {U, } (so D is given by local data {U, f}

6 The Chern connection and the first Chern class 28

and = f/f), then if s := s|U then s = s . Hencess

= =ff,

and thus by definition the function g defined by g|U = fs is a well defined meromorphic functionon X. Then D = (g), since the s are nowhere vanishing.

Next, if : Z X is a holomorphic map between complex manifolds and D a divisor on Xsuch that D is a well defined divisor on Z. Then if D has local data {U, f} then [D] hascocycle

{1(U), f/f

}and [D] has cocycle

{1(U), f/f

}which are visibily the

same.Finally if D Div(X) is given by local data {U, f} then the {f} define a meromorphic

section k of [D] such that (k) = D, where k := f. Conversely if L is given by cocylce{U, }and k is any non-trivial global meromorphic section of L then k/k = , and hence L = [(k)].Thus the image of [] is precisely the isomorphism classes of line bundles admitting non-trivialglobal meromorphic sections. This completes the proof. I

To sum up, we have shown that L is the line bundle associated to some divisor on X if andonly if it has a gloval meromorphic section that is not identically zero; it is the line bundle of aneffective disivosr if and only if it has a nontrivial global holomorphic section, and is trivial if andonly if this nontrivial global holomorphic section never vanishes.

We conclude this section with the following remark. Proposition 5.9 showed that a divisor ona complex manifold D can either be given as a sum

i aiYi or by local data {U, f}. In general

these two types of divisors are not equivalent; the former are Weil divisors and the latter areCartier divisors, and, for instance, they do not necessarily define the same class of objects on acomplex algebraic variety.

6 The Chern connection and the first Chern class

6.1 DefinitionsLet E be a smooth complex vector bundle of rank m over a compact complex manifold X. AHermitian metric , on E is an assignment of Hermitian inner product , p to each fibre Epof X. We say that (E, , ) is a Hermitian vector bundle over X. An easy argument usingpartitions of unity shows that any smooth complex vector bundle admits a Hermitian metric. Alocal frame e = {e1, . . . , em} of E is unitary if ei, ej = ij . Such a frame exists near any givenpoint p, since the Gram-Schmidt orthogonalisation process allows one to find m local sectionswhich form an orthonormal basis for Eq at all points q near p.

6.2 DefinitionLet E X be a rank m complex vector bundle. We let ArC = ArC,E denote the sheaf defined by

ArC (U) = (U,r (T C(X))C E) .

Similarly we let Ap,q denote the sheaf defined by

Ap,q(U) = (U,p,q(X)C E) .

By definition A0,0 := A0C.

6.3 Recap on conncetionsSimilarly to the real case we define a connection D on E to be a C-linear sheaf homomorphismD : A0C A1C such that D(fs) = df s + fDs for a local compled valued C function f and alocal smooth section s. As in the real case, a connection extends uniquely to define a covariant


derivative dE : ArC ArC defined as follows: if is a local complex r-form and s a local section ofE then

dE ( s) = d s+ (1)r Ds.The cuvature R of a connection D is simply the map dE dE : A0 A1. A connection is givenlocally with respect to a local frame e = {e1, . . . , em} defined over U X by a matrix =

[ij]

of complex 1-forms, ij 1C(U), and the curvature is given by a matrix =[ij]of complex

2-forms, ij 2C(U) defined by

Dej = ijei,

Rej = ijei,

and = d + .

If D is a connection on (E, , ) we say that D is a unitary connection if D is compatiblewith , in the sense that if s1, s2 are any two local sections then

d s1, s2 = Ds1, s2+ s1, Ds2 .

Equialently, D is is a unitary connection if for any unitary frame e = {e1 , . . . , em) of E theconnection matrix is a skew Hermitian matrix, that is, t = .

6.4 DefinitionLet pi : E X be a complex vector bundle of rank m. A partial connection on E is a C-linearsheaf homomorphism

E : A0,0 A0,1

satisfying(fs) = f s+ fEs

for a local smooth complex valued function f and a local smooth section s. Similarly we can extenda partial connection E to define a partial covariant derivative, that is, sheaf homomorphismE : Ap,q Ap,q+1 defined as follows: if is a local (p, q)-form and s a local smooth section of Ethen we define

E ( s) := s+ (1)p+q Es.For p+ q = 0 this is just the standard definition, which shows that this extension is well defined,that is, E( fs) = E(f s). Indeed,

E ( fs) = fs+ (1)p+q E(fs)= f s+ (1)p+q (f s+ fEs)= (f) s+ (1)p+qf Es= E (f s) .

Moreover E satisfies the following generalised Leibniz rule: if is any local section of Ap,q and any local (r, t)-form then

E ( ) = + (1)r+t E.

Indeed, without loss of generality = s where is a local (p, q)-form and s is a local sectionof E. Then

E ( ) = E ( s)= ( ) s+ (1)p+q+r+t ( ) Es= + (1)r+t ( s+ (1)p+q ( ) Es)= + (1)r+t E.


One easily checks that the operator E E : A0,0 A0,2 is linear over the smooth comple valuedfunctions, that is,

E E(fs) = fE E(s),and thus E E is induced by a bundle morphism E 0,2(X) C E and hence may defines aglobal section of A0,2End(E)(X) =

(X,0,2(X)C End(E)

).

6.5 An alternative definition of a holomorphic vector bundleWe call a partial connection E a holomorphic structure if E E = 0. We define a holo-morphic vector bundle to be a pair

(E, E

)where E X is a smooth complex vector bundle

and E a holomorphic structure on E (contrast Definition 4.1). We then define a holomorphicsection of E to be a section s such that Es = 0.

J Let us check that this is equivalent to Definition 4.1. Suppose pi : E X is a holomorphicvector bundle of rank m in the sense of Definition 4.1. Let e = {e1, . . . , em} be a holomorphic localframe over U X, so a local section Ap,q(U) can be written as

= i ei,

with i p,q(U). Define E byE = i ei. (17)

If we can show this is well defined then this clearly a holomorphic structure on E such that theholomorphic sections are then precisely the (standard) holomorphic sections of E. To check it iswell defined, suppose e = {e1, . . . , em} is another holomorphic local trivialisation over U . Thenwe have

ei = ji ej

for some holomorphic map =[ij]

: U GL(m,C). We obtain a different operator E usingthe frame e, but luckily

E (i ei

)= E

(i ji ej

)= E

(iji ej

)=

(iji

) ej

()= i ji ej= i ji ej= i ei= E

(i ei

),

and so the definitions coincide - note that we used the fact that the ji are holomorphic in ().Thus we can associate a well defined holomorphic structure E to E.

Conversely if(E, E

)is a holomorphic vector bundle in the new sense then the Koszul-

Malgrange integrability theorem states that we can find a cocycle {U, } of E such thatE is given locally by (17). Working through the above computation backwards, () then showsus that = 0, and thus the transition functions are holomorphic, and so E admits thestructure of a holomorphic vector bundle in the sense of Definition 4.1. I

6.6 DefinitionLet pi : E X be a smooth complex Hermitian vector bundle and D a connection on E. We canwrite D = D1,0 + D0,1, where D1,0 : A0,0 A1,0 and D0,1 : A0,0 A0,1 are the components ofthe C-linear sheaf homomorphism D : A0C A1C. Equivalently, in any local trivialisation U wehave = 1,0 + 0,1 where 1,0 is a matrix of (1, 0)-forms and 0,1 is a matrix of (0, 1)-forms.


This decomposition does not require E to be holomorphic, but if E is holomorphic then wehave a holomorphic structure E on E defined by (17). We thus can say that a connection D on aholomorphic vector bundle is compatible with the complex structure if D0,1 = E whereE is defined by (17).

6.7 PropositionLet E X be a holomorphic complex Hermitian vector bundle of rank m. Then there exists aunique unitary connection D on E that is compatible with the complex structure. We call D theChern connection on E.

J Let e = {e1, . . . , em} be a holomorphic frame for E, and h =[hij]

= [ei, ej]. If such aconnection D exists then its matrix with respect to e must be of type (1, 0) and satisfy

dh = h + t h.

Thush = h ,h = t h,

from which if follows that = h1 h. (18)

This proves uniqueness, and also proves existence locally. Thus we can find local connections = h1 h satisfying the requirements, which must agree on the overlaps by uniqueness. Thusthe local connections glue together to define the desired global connection D. I

We also observe that

= (h1 h)

= h1 h h1 h= ,

and thus

= d + = + + = . (19)

Thus 1,1(U).

6.8 Specialising to the case of a line bundleThe discussion becomes somewhat simpler when we specialise to the case of a complex line bundle.Recall in general that if L has cocycle {U, } then if D is a connection on L then if denotesthe connection 1-form (is now a 11 matrix of 1-forms; that is, 1C(U)) of D with respectto the U trivialisation then for U U 6= , and are related by

= 1 d + 1 . (20)

But if L is a line bundle then (p) GL(1,C) = C for all p UU ; since C is commutative(20) reduces to

= 1 d + .Moreover since the curvature 2-forms of D are related by

= 1 = ,


we see that the curvature is a well defined 2-form 2C(X). Moreover as

= d + = dsince = 0 as is a 1-form, we see that is locally exact, and hence locally closed. Thus is globally closed, as closedness is a local property, but in general is not globally exact, sinceexactness is not a local property. Thus defines a cohomology class [] H2C,dR (X)

If D and D are two connections on L with curvature 1-forms , then if := thensince on U U ,

= =

(1 d +

)(1 d +

)= ,= ,

we see that the glue to give a well defined 1-form 1C(X). Thus if and are thecorresponding curvature 2-forms then

= d,and hence the cohomology class [] is independent of the choice of connection D on L.

Finally if now D denotes the Chern connection on a holomorphic line bundle L then (18) and(19) tell us the following: if e is a non-vanishing holomorphic section on U then we now have

h = e, e = |e|2 ,

and so =

hh

= log |e|2 ,

and thus

= = log |e|2 .=

i

2ddc log |e|2 . (21)

The argument above for the Chern connection, together with the fact that [] is independent ofthe choice of connection shows that [] actually defines a cohomology class in H1,1(X).

6.9 DefinitionWe can now define the first Chern class c1(L) of a smooth complex line bundle L over a compactcomplex manifold X. Choose a unitary connection D on L; then since L has rank 1, the conditionon is that is purely imaginary. Thus i is a real form, and we define

c1(L) :=[i2pi

] H2dR (XR) .

We will not prove the following theorem; its proof is essentially algebraic topology, and thusfalls out of the remit of this course.

6.10 TheoremLet X be a complex manifold. Then if L is a complex line bundle over L then c1(L) determines Lup to complex vector bundle isomorphism.

In particular, this theorem shows that if a line bundle L has c1(L) = 0 then L is differentiablytrivial, that is, L admits a smooth global non-vanishing section. If L is a holomorphic line bundle


this does not in general necessarily imply that L is also holomorphically trivial, that is, thatthere exists a holomorphic non-vanishing section. At the end of this Chapter (see Proposition6.18) however we will see that in the special case X = CPn then these concepts coincide; thatis, if a holomorphic line bundle L over CPnis differentiably trivial then it is also holomorphicallytrivial.

6.11 LemmaLet L and L be two smooth complex line bundles over a compact complex manifold X. Thenc1(L L) = c1(L) + c1(L).

J If D and D are connections on L and L respectively then we can define a connection D onL L by

D (s s) = Ds s + sDs.If dLL

denotes the corresponding covariant deriviative, and R the curvature of D and Rthe

curvature of Dthen the curvature R of D is computed as

R(s s) = dLL(dLL

(s s)

)= Rs s DsDs +DsDd + sRs= (R+R) (s s).

Since we can compute c1 (L L) with respect to a connection on LL of our choice, using D itthus follows that c1 (L L) = c1(L) + c1(L) as claimed. I

6.12 CorollaryLet L be a complex line bundle over a complex manifold X. Then c1 (L) = c1(L).

J Immediate from the above, since the trivial bundle has a flat connection. I

6.13 Divisors and the first Chern classLet Y X be an irreducible analytic hypersurface, where Xn is a compact complex manifold.Let Y H2dR (XR) denote the Poincar dual class to the fundamental cycle [Y ] H2n2 (XR,Z).Next, given any divisor D =

i aiYi on X, we define D =

i aiYi H2dR (XR).

6.14 TheoremLet Xn be a compact complex manifold and D Div(X). Then c1 ([D]) = D H2dR (XR).

J It is sufficient to show that if 2n2 (XR) is a closed (real) form and is curvature2-form of the Chern connection on [D] then

i

2pi

X

=Ni=1

ai

Yi

,

where D =Ni=1 aiYi. Since both D 7 c1([D]) and D 7 D are both homomorphisms Div(X)

H2dR (XR), it is sufficient to consider the case where D = Y is an irreducible analytic subvariety.Let s be a holomorphic section of [Y ], so Y = (s). Now pick a Hermitian product on X and

defineX() := {p X| |s(p)| > } .

Then we have by (21) that

i

2

X

= 14

lim0

X()

(ddc log |s|2

) .


Since d(dc log |s|2

)=(ddc log |s|2

) since is closed we have by Stokes Theorem

X()

(ddc log |s|2

) =

X()

dc log |s|2 .

Now if U is a holomorphic trivialising neighborhood with local non-vanishing holomorphic forme and s O(U) is defined (so s|U = se), and h = e, e

|s|2 = |s|2 h = s s hon U (X\X()). Thus

dc log |s|2 = i ( ) log (s s h)= i

( log s log s +

( ) log h) ,

since s is holomorphic. Now since vol(X()) 0 as 0 and dc log h is bounded below bysome > 0 on U, dc log h is bounded away from Y and thus

lim0

X()U

dc log h 0.

Next, since is a real form,X()U

log s =X()U

log s ,

and thus

14pi

lim0

X()U

dc log |s|2 = i2pi

lim0

(Im

(X()U

log s ))

.

Hence the result will follow if we can show

lim0

X()U

log s = 2piiY U

. (22)

This is a purely local question, so we can reduce to the case X = P a polydisc about the originin Cn and Y = {z1 = 0} P . Then Y = (z1) and X() = {h (z1) = } and we want to show

lim0

{h(z1)=}

log z1 = 2pii{{z1=0}

. (23)

Now log z1 = dz1

z1 and if we write

= fdz2 dzn dz2 dzn + dz1 1 + dz1 2for f holomorphic then dz

1

z1 dz1 1 = 0 and dz1

z1 dz1 2 doesnt contribute to the integralover

{h(z1)

= }. Thus

{z1=0}

={z1=0}

f(0, z2, . . . , zn

)dz2 dzn dz2 dzn. (24)

Now {h(z1)=}

dz1

z1 =

{h(z1)=}

fdz1

z1 dz2 dzn dz2 dzn

and so

lim0

{h(z1)=}

dz1

z1 =

(lim0

|z1|=/h

fdz1

z1

) P

dz2 dzn dz2 dzn

= 2pii{z1=0}

f(0, z2, . . . , zn

)dz2 dzn dz2 dzn,

which is the same as (24); this verifies (23) and hence (22), and thus completes the proof. I


6.15 CorollaryFor any divisor D, the first Chern class c1([D]) lies in the image of the map H2 (XR,Z) H2 (XR,R) = H2dR (XR).

J Immediate as the fundamental cycle [Y ] lies in H2n2 (XR,Z). I

6.16 The Picard group of a compact connected Riemann surface Let denote a compact connected Riemann surface. A divisor D on is just a finite formal sumof points, D =

ni=1 aiPi where Pi is a point.

We define the degree of D to beni=1 ai. Equivalently the degree of D is its fundamental class

[D] H0 (R,Z). This gives us a map deg : Div() H0 (R,Z) which is evidently surjective. IfP is a point then [P ] H0 (R,Z) gives rise to the Poincar dual P H2 (R,Z). Under themap

H0 (R,Z) H0 (R,R) H0dR (R)we may think of P 2 (R) as a closed form which we may assume satisfies

RP = 1. (25)

Similarly we define the degree of a line bundle L on by

deg(L) := c1(L), [R] ,that is, deg(L) = c1(L) under the isomorphism H2 (R,Z) = Z given by the natural orientationon . Thus if L = [D] where D =

ni=1 aiPi then

deg ([D]) =12pii

=ni=1

ai

S

Pi = deg(D),

where we are using Theorem 6.12 and (25).Thus we also have a homomorphism deg : Pic() Z which is surjective.

6.17 Specialising to the case = CP 1

The non-constant holomorphic maps : CP 1 CP 1 are precisely the non-constant rationalfunctions. Moreover thinking of a rational function as a map C {} C {} then thenumber of zeros of is the same as the number of poles of , counted with multiplicity andincluding zeros and poles at . Moreover if P1, . . . , Pn are any distinct points in C {} anda1, . . . , an Z\{0} such that

ni=1 ai = 0 then there exists a rational function such that

has zeros (ai > 0) or poles (ai < 0) precisely at the points Pi, with the multiplicity given by |ai|;namely set

(z) =

ai>0

(z Pi)aiai


section k of LO(m), then we can obtain a non-trivial global meromorphic section of L from kand sm.

So suppose c1(L) = 0. Then there exists (by Theorem 6.10) a global never zero smoothsection s of L. If L has cocycle {U, } then s is given by smooth functions s C(U,C)with s = s on UU . Since the are holomorphic, it follows s/s O(UU). Sincethe s are never zero, we can write s = ef where f C(U,C), and then ff O(UU).Thus the local -closed forms f agree on the overlaps and so patch together to define a -closedform 0,1 (CPn). We shall prove in Chapter 8 that H0,1 (CPn) = 0 using Hodge theory(note that in the case n = 1 this can be deduced directly from the -Poincar Lemma (Theorem2.10). Hence there exists a smooth complex valued function f on CPn such that f = . Then ift := eff then t = 0, so t is holomorphic on U, and t = t on UU . Thus the t gluetogether to define a global never zero holomorphic section of L, and thus L is holomorphicallytrivial. But then of course L trivially admits a non-trivial global meromorphic section; thiscompletes the proof. I

6.19 CorollaryPic(CP 1) = Z where the isomorphism sends L 7 deg(L).

J Combining Section 6.17 with Proposition 6.18 and Theorem 5.13 shows that

Div(CP 1

)/{prinicpal divisors} = Pic (CP 1)

and that deg : Div(CP 1

)/{prinicpal divisors} Z is an isomorphism. I

In fact, we can get even more information about the structure of Pic(CP 1

).

6.20 PropositionRecall the line bundles O(m) over CPn defined in Section 4.6. Then deg (O(m)) = m.

In particular, specialising to the case n = 1 we see that thus up to isomorphism, the onlyholomorphic line bundles over CP 1 are the O(m) for m Z.

J Let Ha := {w CPn | a w = 0}where a Cn+1\{0}. Then Ha is a hyperplane in CPn;moreover every hyperplane is of this form. Given a, b Cn+1\{0}, k := awbw is a non-trivial globalmeromorphic section with (k) = Ha Hb. Thus Ha Hb is prinicpal, and thus the line bundles[Ha] and [Hb] conicide. Thus all hyperplanes define the same line bundle, which we will denote by[H].

In particular take H ={w0 = 0

}. Then with Ui =

{wi 6= 0}as before we can take as a local

defining function fi for H over UI the function fi = z0/zi. It thus follows that [H] has cocycle

{Ui, ij} where ij = fi/fj = zj/zi. But it precisely the cocycle of O(1) = O(1) (see Lemma 4.6)and thus we conclude

O(1) = [H](this is the reason O(1) is called the hyperplane bundle). Note clearly that by definitiondeg ([H]) = 1, and hence deg (O(1)) = 1. Then finally since [] is a homomorphism we have

O(m) = O(1)m = [H]m = [mH],and thus deg(O(m)) = deg ([mH]) = m. I

More generally we can use this proof to show that if P is a homogen

complexiii

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