Part III- Hodge Theory Lecture Notes

Anne-Sophie KALOGHIROS

These lecture notes will aim at presenting and explaining some specialstructures that exist on the cohomology of Kahler manifolds and to discusssome of the properties and consequences of these structures from the pointof view of complex algebraic geometry. We will concentrate on nonsingularprojective varieties, a special case of compact Kahler manifolds.

Overview

A complex manifold is a topological space that is locally modelled on openpolydiscs of Cn and equipped with holomorphic transition functions. Thenatural isomorphism Cn ' R2n endows X with the structure of a differen-tiable manifold of dimension dimRX = 2 dimCX. These two structures onthe underlying topological space turn out to behave quite differenttly in gen-eral; this reflects the fact that holomorphic function theory is in some sensemuch more “rigid” than differentiable function theory. If X is a complexmanifold, the tangent bundle of the associated differentiable manifold TX,Rcan be complexified TX,C = TX,R⊗C and equipped with a Hermitian metrich = g − iω. This metric is inherited from the identification of Tx,X,C witha complex vector space at each point x ∈ X, and it varies smoothly withx. The complex manifold X is Kahler when the metric ω is closed for theexterior differential; this condition ensures compatibility between the com-plex structure of X and its differentiable structure. Nonsingular projectivevarieties are Kahler, and in fact Kahler manifolds can be thought of as adifferential geomeric generalisation of these. However, this analogy shouldbe taken with a pinch of salt; for instance, a Kahler manifold does not ingeneral have any complex submanifold. Note that it is in general difficultto decide whether a given complex manifold is Kahler, or even to constructnon projective Kahler manifolds.

Let X be a complex manifold and TX,R its tangent space (when viewedas a differentiable manifold). At every point x ∈ X, Tx,X,R is equipped witha complex structure Jx, i.e. an endomorphism of Tx,X,R such that J2

x = −id,

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which can be extended to the complexified tangent space Tx,X,R⊗C = Tx,X,C,and induce a decomposition:

Tx,X,R ⊗ C = T 1,0x,X ⊕ T

0,1x,X

where T 1,0x,X = u ∈ Tx,X,C|Jxu = iu and T 0,1

x,X = u ∈ Tx,X,C|Jxu =−iu. This direct sum decomposition holds at the bundle level, and dualisesto a similar decomposition on the cotangent bundle, the vector bundle ofdifferential 1-forms Ω1

X,C = (TX,C)∗, and by extension, on the bundle of

differtial k-forms ΩkX , namely:

ΩkX,C = Ωk

X ⊗ C =⊕p+q=k

Ωp.qX ,

where we define Ωp,qX =

∧p Ω1,0X ⊗

∧q Ω0,1X . This decomposition satisfies the

Hodge symmetry, i.e. Ωp,qX = Ωq,p

X , where complex conjugation acts naturallyon Ωk

X,C = ΩkX,R ⊗ C.

Let C∞(X,ΩkX,C) denote the space of complex differential forms of degree

of degree k on X– C∞-sections of ΩkX,C– and denote d : C∞(X,Ωk

X,C) →C∞(X,Ωk+1

X,C) the exterior differential, with d d = 0. Recall that the kthde Rham cohomology group of X are then defined as:

Hk(X,C) =ker(d : C∞(X,Ωk

X,C)→ C∞(X,Ωk+1X,C)

im(d : C∞(X,Ωk−1X,C)→ C∞(X,Ωk

X,C).

Theorem 0.1 (Hodge Decomposition). Let X be a compact Kahler mani-fold. If Hp,q(X) ⊂ Hk(X,C) is the set of De Rham cohomology classes thatcan be represented by a closed form α of type (p, q) at every point x ∈ X,then we have a decomposition:

Hk(X,C) =⊕p+q=k

Hp,q(X),

and the summands satisfy Hodge symmetry Hp,q(X) = Hq,p(X).

Remark 0.2. The Hodge decomposition theorem states that on a Kahlermanifold, the decomposition of degree k differential forms into forms of type(p, q) with p+ q = k descends to the De Rham cohomology.

Remark 0.3. Note that in the statement of the Hodge Decomposition, wehave written the = sign between various cohomology rather than '; this isbecause the isomorphisms in question will be shown to be canonical.

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Remark 0.4. Even though the Kahler condition is crucial in order to proveTheorem 0.1, we will see that the Decomposition itself does not depend onthe choice of Kahler metric, it only depends on the complex structure.

The principle behind the proof of the Hodge Decomposition Theoremis that each De Rham cohomology class has a unique representative thatis a harmonic form for an elliptic differential operator, the Laplacian ∆d.The Kahler hypothesis is crucial in the proof of this identification and alsoprovides a decomposition of harmonic forms into forms of type (p, q).

Another consequence of this principle of representation of cohomologyclasses by harmonic forms is the Lefschetz Decomposition. As has beenmentioned above, if X is a complex manifold of dimension n, X can beendowed with a Hermitian form that arises from its differentiable structureh = g− iω. When X is Kahler, the form ω is a representative of a (cohomol-ogy) class [ω] in H2(X,R). The exterior product with the class [ω] definesan operator L : Hk(X,R)→ Hk+2(X,R).

Theorem 0.5 (Hard Lefschetz Theorem). Let X be a compact Kahler man-ifold. For every k ≤ n = dimCX, the map

Ln−k : Hk(X,R)→ H2n−k(X,R)

is an isomorphism. In particular, L : Hk(X,R) → Hk+2(X,R) is injectivefor k < n.

The primitive cohomology is then

Hk(X,R)prim = ker(Ln−k+1 : Hk(X,R)→ H2n−k+2(X,R)).

Theorem 0.6 (Lefschetz Decomposition). Let X be a compact Kahler man-ifold. The natural map

i :⊕

k−2r≥0

Hk−2r(X,R)prim → Hk(X,R)

(αr) → ΣrLrαr

is an isomorphism for k ≤ n

The existence of these decompositions on the De Rham cohomologygroups of a compact Kahler manifold has important consequences. For in-stance, define the Betti numbers by bk(X) = dimCH

k(X,C) and the Hodgenumbers by hp,q(X) = dimCH

p,q(X). By Theorem 0.5, the odd (resp even)

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Betti numbers b2k−1(X) (resp. b2k(X)) increase with k for 2k − 1 ≤ n(resp. 2k ≤ n). Theorem 0.1 shows that

bk(X) = Σp+q=khp,q(X),

and by Theorems 0.1 and 0.6, we have:

hp,q(X) = hq,p(X) = hn−p,n−q(X) = hn−q,n−p(X),

so that, in particular, the odd Betti numbers are even.

The Hodge Decomposition has further reaching consequences when itis combined with the integral structure on the cohomology Hk(X,Z). Itwould be nonsensical to consider the De Rham complex (i.e. degree kdifferential forms) to compute Hk(X,Z)- we will use the methods of sheafcohomology to compute Hk(X,Z). Using the language of sheaves, we willidentify the summand Hp,q(X) in Theorem 0.1 as the Dolbeault cohomologygroups Hq(X,Ωp

X)–the q-th cohomology groups of X with values in the sheafof holomorphic differential forms of degree p (this identification is possiblebecause of the Kahler hypothesis).

We will formalize these results on the cohomology of compact Kahlermanifolds by introducing the notion of Hodge Structures. An integral Hodgestructure of weight k is an abelian group of finite type HZ and a Hodgedecomposition

HC = HZ ⊗ C =⊕p+q=k

Hp,q,

with Hp,q = Hq,p. We have seen that a Hodge structure exists on the degreek cohomology of a Kahler manifold.

Here is an example of application of these Hodge Structures. If X is anonsingular projective variety, (H1(X,Z), H1(X,C) = H1,0(X)⊕H0,1(X))is a weight 1 Hodge structure. To this Hodge structure, we may associatea complex torus T = Ck/Γ = H0,1(X)∗/H1(X,Z), where k = b1(X) and Γis a lattice of rank 2k. This complex torus is the Picard variety of X andwe will see that it parametrises the holomorphic line bundles L on X whichhave trivial Chern class c1(L).

As I mentioned in Remark 0.4, the Hodge structure only depends onthe complex structure and not on the differentiable structure, that is on thechoice of a Kahler metric. We can ask how these Hodge Structures varywith the complex structure. These questions amount to studying varyingdecompositions on a fixed vector space. Indeed, the De Rham cohomol-ogy groups are invariant under diffeomorphism (differentiable isomorphism),

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they are even topological invariants. In particular, when the complex struc-ture varies and the differentiable structure is “fixed”, these groups will notchange, but the decomposition into direct summands will. We will showhow to construct a period domain D that “parametrises” deformations ofHodge Structures when the complex structure varies (that is when HZ = Γ0

and HC = Γ0 ⊗ C are fixed, but the decomposition on HC varies).In fact, we will see that these questions are related to the study of “fam-

ilies” of compact complex manifolds. If X → B is a “family of compactcomplex manifolds” over a contractible base B, then the fibres Xt, t ∈ Bare all diffeomorphic to the central fibre X0. We will see that in some cases,there is a “universal” family of deformations X → B of the central fibre X0.The period map P : B → D associates to t ∈ B the Hodge structure onHk(Xt,C) ' Hk(X0,C). A fundamental result due to Griffiths is:

Theorem 0.7. The period map is holomorphic.

In fact, in nice cases, this period map is even an embedding or a submer-sion, and understanding how the Hodge Structure varies locally gives muchinformation about the deformations of the manifold itself.

1 Complex manifolds

In these notes, I will assume some familiarity with basic notions of differen-tial and algebraic geometry. Let U ⊂ Cn be an open subset and f : U → Ca complex valued function. The function f is differentiable if after someidentification Cn ' R2n and C ' R2, the induced function f : R2n → R2 isdifferentiable. This is independent of the choice of identification. In thesenotes, I always use the term differentiable for C∞, but most of the state-ments will hold under weaker assumptions.

1.1 Local Theory: holomorphic functions of several variables

The reader who is not familiar with these notions should read [Voi02, Ch.1]or [Huy05, Section I.1].

Fix a standard system of coordinates (z1, · · · , zn) on U ⊂ Cn, and letxj = <zj and yj = =zj be the canonical linear coordinates of R2n.

Definition 1.1. Let U ⊂ Cn be an open set and f : U → C a differentiablefunction. The function f is holomorphic at ω = (ω1, · · · , ωn) ∈ U if for allj = 1, · · · , n, the function

zj 7→ f(ω1, · · · , ωj−1, zj , ωj+1, · · · , ωn)

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is holomorphic at ωj , that is if

∂f

∂zj(ω) :=

1

2(∂f

∂xj+ i

∂f

∂yj)(ω) = 0.

The function f is holomorphic on U if f is holomorphic at ω for all ω ∈ U .When f : U → Cm is a differentiable function, f is holomorphic if each

fi : U → C is holomorphic, where f = (f1, · · · , fm).

Lemma 1.2. If f : U → C is holomorphic and does not vanish on U , 1/fis holomorphic on U . If f, g are holomorphic maps U → C, fg, f + g andg f (when it is defined) are holomorphic on U .

Proof.

Exercise 1.3. Show that f is holomorphic at ω ∈ U precisely when theR-linear application dfω : Cn → C is C-linear.

Let u = <f : U → R and v = =f : U → R be the real and imaginaryparts of f . Show that f is holomorphic on U if and only if for all j = 1, · · · , n,u and v satisfy the Cauchy-Riemann Equations:

∂u

∂xj=

∂v

∂yjand

∂u

∂yj= − ∂v

∂xj.

Definition 1.4. Let ω = (ω1, · · · , ωn) ∈ U be a point andR = (R1, · · · , Rn) ∈(R∗+)n. The polydisc around ω with multiradius R is:

D(ω,R) = (z1, · · · , zn) ∈ Cn||zj − ωj | < Rj , j = 1, · · · , n.

Theorem 1.5. [Voi02, 1.17] Let U ⊂ Cn be an open subset and f : U → Cbe a differentiable map. The following are equivalent:

1. f is holomorphic at ω for all ω ∈ U ,

2. For all ω ∈ U there is a polydisc D(ω,R) ⊂ U such that f admitsa power series expansion f(z + ω) = ΣIαIz

I for multi-indices I =(i1, · · · , in) ∈ Nn that converges absolutely for ω + z ∈ D.

3. If D = D(ω,R) is a polydisc contained in U , for all z ∈ D,

f(z) = (1

2iπ)n∫|ζj−ωj |=Rj

f(ζ)dζ1

ζ1 − ω1∧ · · · ∧ dζn

ζn − ωn,

where the integral is taken over a product of circles, with the orienta-tion that is the product of the natural orientations.

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Definition 1.6. Let U ⊂ Cn be an open set. A holomorphic map f : U →Cn is locally biholomorphic at ω ∈ U if there is a neighbourhood V of ωwith V ⊂ U such that f|V is bijective onto f(V ) and f−1

|V is holomorphic.

Definition 1.7. Let U ⊂ Cn be an open subset, and f : U → Cm be aholomorphic map. The (complex) Jacobian of f at ω ∈ U is the matrix

Jf (ω) =(∂fk∂zj

(ω))

1≤k≤m,1≤j≤n.

The point ω ∈ U is regular if Jf (ω) is surjective, f(ω) is a regular value ifevery point z ∈ f−1(f(ω)) is regular.

Exercise 1.8. Let U ⊂ Cn be an open subset and f : U → Cn a holomor-phic map. Show that f is locally biholomorphic at ω ∈ U if and only ifdet Jf (ω) 6= 0.

Finally, the following result can be extracted from [Huy05, 1.10,1.11], andshows that a holomorphic map whose Jacobian matrix has locally constantrank locally has a canonical representation.

Theorem 1.9. Let U ⊂ Cn be an open subset and f : U → Cm a holo-morphic map. Let ω ∈ U be a point such that rk Jf (z) = k for all z in aneighbourhood of ω. There are open neighbourhoods V of ω ∈ U and W off(ω) in Cm, and biholomorphic maps ϕ : Dn → V and ψ : W → Dm suchthat the composition ψ f ϕ : Dn → Dm is given by

(z1, · · · , zn) 7→ (z1, · · · , zk, 0, · · · , 0).

1.2 Complex manifolds: definitions and first examples

Definition 1.10. A complex manifold of dimension n is a connected Haus-dorff topological space X equipped with a complex atlas (Ui), φi, where(Ui)i∈I is a countable covering by open subsets, and each φi : Ui → Vi ⊂ Cnis a homeomorphism from Ui onto an open subset of Cn, and for all i, j ∈ I,the transition functions

φj φ−1i : φi(Ui ∩ Uj)→ φj(Ui ∩ Uj)

are biholomorphic.Two complex atlases are equivalent if their union defines a complex atlas.

There is a maximal atlas equivalent to any given complex atlas on X, wesay that such a maximal atlas is a complex structure on X.

A complex manifold X is compact if its underlying topological space iscompact.

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Recall that a smooth manifold M of dimension m is a topological spaceM and a maximal differentiable atlas (Wi)i∈I , ψi∈I , where (Ui)i∈I is acountable covering by open subsets, and each ψi : Wi → Rm is a homeomor-phism from Wi onto an open subset of Rm, and for all i, j ∈ I, the transitionfunctions

ψj ψ−1i : ψi(Wi ∩Wj)→ ψj(Wi ∩Wj)

are diffeomorphic.In particular, every complex manifold of dimension n has a natural struc-

ture of smooth manifold of dimension 2n.

Remark 1.11. Even though the definitions of complex and smooth man-ifolds are very similar, they have crucial differences. For instance, while adifferentiable manifold Mm can always be covered by open subsets that arediffeomorphic to Rm, a complex manifold Xn cannot in general be coveredby open subsets that are biholomorphic to Cn. For instance, take X tobe the unit disc D ⊂ C, then Liouville’s theorem shows that there is nonon-constant holomorphic map C→ X.

Examples

1. Let U ⊂ Cn be an open subset. Then U is a complex manifold,with complex atlas U, id. More generally, if U ⊂ X is a connectedopen subset of a complex manifold X, then it has a complex structureinduced by that of X.

2. Let V be a complex vector space of dimension n+ 1. Let P(V ) denotethe set of lines through 0 ∈ V , i.e.

P(V ) = l ⊂ V |l is a subspace of dimension 1 = Gr(1, V ),

then P(V ) is a complex manifold of dimension n (when V = Cn+1,P(V ) = CPn). For every point v ∈ V \ 0, denote [v] = C · v ⊂ Vthe corresponding point of P(V ), conversely, for every point l ∈ P(V ),there is an element v ∈ V \ 0 that is unique up to multiplication bya constant λ ∈ C∗ such that [v] = l. We have a surjective map:

π : V \ 0 → P(V ),

which endows P(V ) with the quotient topology defined by π and thestandard topology on V . We endow P(V ) with a standard complexstructure as follows. Choose a C-linear isomorphism V ' Cn+1, and

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for every point v = (v0, · · · , vn), denote [v] = [v0: · · · :vn] the homo-geneous coordinates of [v] ∈ P(V ). For each i = 0, · · · , n define theopen set Ui = [v] = [v0: · · · :vn] ∈ P(V )|vi 6= 0 ⊂ P(V ) and thehomeomorphism φi : Ui → Cn

[v0: · · · :vn] 7→ (v0

vi, · · · , vi

vi, · · · , vn

vi).

The transition functions are

φj φ−1i : (z1, · · · , zn) ∈ Cn|zj 6= 0 → (u1, · · · , un) ∈ Cn|ui 6= 0

(z1, · · · , zn) 7→ (z1

zj, · · · , zj

zj, · · · , zi−1

zj,

1

zj, · · · , zn

zj),

these are biholomorphic.

Remark 1.12. As a differentiable manifold, CPn ' S2n+1/S1 (seeexample sheet 1), in particular CPn is compact.

3. Let Λ ⊂ Cn be a lattice of rank 2n. Denote π : Cn → Cn/Λ thequotient map and X = Cn/Λ the quotient. Then X is a complexmanifold. Endow X with the quotient topology of Cn. If U ⊂ Cn is asmall open subset such that U ∩ (U + (Λ r 0)) = ∅, then U → π(U)is bijective. Covering X by such open subsets gives a complex atlas,whose transition functions are just translations by elements in Λ.

More generally, let X be a complex manifold and Γ ⊂ AutX a sub-group of the group of automorphisms of X that acts properly dis-continuously on X, i.e for any two compact subsets K1,K2 ⊂ X,γ(K1)∩K2 6= ∅ for at most finitely γ ∈ Γ. Assume further that Γ actswithout fixed point–i.e. γ · x 6= x for all x ∈ X and 1 6= γ ∈ Γ–then Xis a complex manifold and X → X/Γ is a locally biholomorphic map.

4. (Affine hypersurfaces) Let f : Cn → C be a holomorphic function suchthat 0 is a regular value. Consider

X = f−1(0) = z ∈ Cn|f(z) = 0.

By Theorem 1.9, there is an open cover X = ∪iUi, open subsetsVi ⊂ Cn−1 and holomorphic maps Vi → Cn inducing bijective mapsφi : Ui → Vi. The transition maps φj φ−1

i are biholomorphic, and Xis a complex manifold of dimension n− 1.

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5. (Projective Hypersurfaces) Let f : Cn+1 → Cn be a homogeneous poly-nomial, and assume that 0 ∈ C is a regular value for the induced holo-morphic map f : Cn+1\0 → C. Then, the affine hypersurface f−1(0)is a complex manifold. The projective hypersurface

X = [z0: · · · :zn] ∈ CPn|f(z0, · · · , zn) = 0 ⊂ CPn

is a complex manifold of dimension n−1. The open subsets Ui∩X forman open cover of X, where Ui are the standard charts of CPn definedabove. Using the isomorphism Ui ' Cn above, X ∩ Ui is identifiedwith f−1

i (0), for fi(z1, · · · , zn) = f(z1, · · · , zi−1, 1, zi+1, · · · zn) and bythe previous example, we can find a complex atlas.

Definition 1.13. Let X be a complex manifold of dimension n, equippedwith a complex atlas Ui, φi : Ui → Cni∈I and Y a complex manifold ofdimension m equipped with a complex atlas Wj , ψj : Wj → Cmj∈J . Aholomorphic map f : X → Y is a continuous map such that for all (i, j) ∈I × J ,

ψj f φ−1i : Cn → Cm

is holomorphic. If Y = C, we say that f is a holomorphic function onX. Two complex manifolds X and Y are biholomorphic if there exists aholomorphic homeomorphism f : X → Y .

Definition 1.14. Let X be a complex manifold, define the structure sheafOX as the (pre)sheaf (see Definition 1.50, and the discussion there):

OX(U) = Γ(U,OX) = f : U → C|f is holomorphic,

where U is an open subset of X. The presheaf OX is a sheaf of rings (therestriction morphisms are ring morphisms). If x ∈ X, define

OX,x = limx∈UOX(U),

where the limit is taken over all open subsets that contain x ∈ X.

Remark 1.15. Let X be a complex manifold and (U, φ : U → Cn) be aholomorphic chart. By definition, if x ∈ U is such that φ(x) = 0 ∈ Cn, thereis a natural identification OCn,0 ' OX,x.

Remark 1.16. When X is a differentiable manifold, we define the sheaf ofdifferentiable functions AX = C∞(X) on X as the (pre)sheaf U 7→ AX(U) =f : U → R|f is differentiable.

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Proposition 1.17. Let X be a compact connected complex manifold, thenΓ(X,OX) = C, i.e. every global holomorphic function is constant.

Proof. Let f : X → C be a holomorphic function. Since X is compact andf is continuous, f admits a maximum at a point x ∈ X. If (U, φ) is aholomorphic chart with x ∈ U , f φ−1 is locally constant by the maximumprinciple, and hence constant because X is connected.

Remark 1.18. Using Hartog’s Theorem (cf Example Sheet 1), if X is acomplex manifold of dimension at least 2, Γ(X \ x,OX) = Γ(X,OX) forall x ∈ X, so that if X is compact and connected Γ(X \ x,OX) = C.

Remark 1.19. In a sense, complex manifolds define a much more “rigid”structure than smooth manifolds; this reflects the properties of holomorphicfunctions vs. differentiable functions. We may give an equivalent definitionof complex (resp. differentiable) manifolds that is closer to the spirit ofalgebraic geometry: a complex (resp. differentiable) manifold is a ringedspace (X,OX), where X is a topological space and OX is the structuresheaf, whose sections we define to be the holomorphic (resp. differentiable)ones.

Definition 1.20. Let X and Y be complex manifolds. A holomorphic mapf : X → Y is a submersion (resp. an immersion) if for all x ∈ X, thereis a neighbourhood U(x) of x such that rk Jf (z) = dimY (resp rk Jf (z) =dimX) for all z ∈ U(x). The map f is an embedding if it is an immersionand if f is an homeomorphism from X onto f(X).

Remark 1.21. The rank of the Jacobian matrix does not depend on thechoice of coordinate charts.

Definition 1.22. Let X be a complex manifold of dimension n and Y ⊂ Xbe a closed subset. The subset Y is a closed submanifold of X of codimensionk if for all x ∈ Y , there is an open neighbourhood U ⊂ X of x and aholomorphic submersion f : U → Dk such that U ∩ Y = f−1(0).

Example 1.23. Let X and Y be complex manifolds of dimension n and m,f : X → Y be a holomorphic map and y ∈ Y such that rkJf (z) = m for allz ∈ f−1(y); f−1(y) is a submanifold of dimension n−m. If f : X → Y is anembedding, then f(X) is a submanifold of Y .

Definition 1.24. A projective manifold is a submanifold X ⊂ CPN suchthat there exist homogeneous polynomials f1, · · · , fk ∈ C[Xo, · · · , XN ] ofdegrees d1, · · · , dk with

X = x ∈ CPN |f1(x) = · · · = fk(x) = 0. (1)

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Remark 1.25. Note that if the Jacobian matrix

J =(∂fj∂zl

)1≤j≤k,0,≤l≤N

has rank k everywhere, X is a submanifold of CPN of codimension k. Moregenerally, (1) defines a submanifold of CPN if the Jacobian matrix has max-imal rank. If the Jacobian matrix does not have maximal rank everywhere,(1) defines a projective algebraic variety and the points where the Jacobianhas less than maximal rank are the singular points of the variety.

Definition 1.26. A projective manifold X ⊂ CPn of dimension m that isdefined by m−n homogeneous polynomials such that the Jacobian has rankn−m in every point is a complete intersection.

Exercise 1.27. Show that

C = [x0: · · · :x3] ∈ P3|x0x3 − x1x2 = x21 − x0x2 = x2

2 − x1x3 = 0

is a submanifold of dimension 1. Is it a complete intersection?

Exercise 1.28. Let f = (f1, · · · , fn) : Cn → Cn be a holomorphic map, and(z1, · · · , zn) the standard coordinates on Cn. Define xk = <zk, yk = =zkand uj = <fj , vj = =fj . The (real) Jacobian of f at a ∈ Cn is

JR(f)(a) =(∂(u1, v1, · · · , un, vn)

∂(x1, y1, · · · , xn, yn)

)(a).

Show that det JR(f)(a) = | det J(f)(a)|2, and deduce that any complex man-ifold is orientable.

1.3 Vector bundles

We will want to distinguish between two different notions:

• complex vector bundles are differentiable vector bundles that havevalues in C, i.e. they have differentiable transition functions,

• holomorphic vector bundles have holomorphic transition functions.

Definition 1.29. Let X be a differentiable manifold. A complex vector bun-dle of rank r over C is a differentiable manifold E endowed with a surjectivemap π : E → X such that:

1. For all x ∈ X, the fibre Ex = π−1(x) ' Cr has the structure of aC-vector space of dimension r,

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2. There is an open cover X = ∪iUi and local trivialisations (Ui, hi),where hi : π

−1(Ui) → Ui × Cr is a diffeomorphism with π|π−1(Ui) =p1 hi and p2 hi : Ex → Cr a C-vector space isomorphism for allx ∈ Ui.

The manifold E is the total space of the vector bundle and X is its basespace. Given two local trivialisations (Ui, hi) and (Uj , hj) of E,

hi h−1j : (Ui ∩ Uj)× Cr → (Ui ∩ Uj)× Cr

induces a differentiable map gi,j : Ui ∩ Uj → GL(r,C), where gi,j(x) = hxi (hxj )−1 is a C-linear automorphism. The gi,j are the transition functions ofE.

Exercise 1.30. Check that the transition functions satisfy the cocycle con-ditions:

gi,j gj,k gk,i = Id on Ui ∩ Uj ∩ Uk, and gi,i = Id on Ui.

Remark 1.31. A complex vector bundle E of rank r is determined uniquelyby the differentiable cocycle Ui, gi,j : Ui ∩Uj → GL(r,C). Let E = tiUi×Cr and define (x, v) ∼ (y, w) if x = y ∈ Ui ∩ Uj , and w = gi,j(x) · v, then

E = E/ ∼.

Example 1.32. Let X be a differentiable manifold of dimension m; ifUi, φi is a differentiable atlas of X, the real tangent bundle TX,R is the vec-tor bundle associated to the cocycle Ui, gi,j = JR(φi φ−1

j ) φj, where JRis the (real) Jacobian. Via the inclusion GL(m,R) ⊂ GL(m,C), the tran-sition functions gi,j : Ui Uj → GL(m,C) define a complex vector bundleTX,C = TX,R ⊗ C, the complexified tangent bundle of X.

Definition 1.33. Let X be a complex manifold and π : E → X a com-plex vector bundle associated to a cocycle Ui, gi,j : Ui Uj → GL(r,C).The vector bundle E is holomorphic if E is a complex manifold and if thetransition functions gi,j are holomorphic.

Remark 1.34. A natural question is to ask in how many (non-isomorphic)ways a given complex vector bundle can be seen as a holomorphic vectorbundle. This is in general a non-trivial question; In some cases, no holomor-phic structure exists, but there can also be several different holomorphicstructures. This is already the case for line bundles: on a complex torusCn/Γ, the trivial complex bundle of rank 1 admits many holomorphic struc-tures.

13

The first example of holomorphic vector bundle is of course that of thetrivial vector bundle E = X × C.

Example 1.35. (The holomorphic tangent bundle) Let X be a complexmanifold of dimension n endowed with a complex atlas Ui, φi : Ui → Vi ⊂Cn. Then, TX is the holomorphic bundle of rank n associated to the co-cycle Ui, gi,j = J(φi φ−1

j ) φj : (Ui ∩ Uj) → GL(r,C), where J is thecomplex (holomorphic) Jacobian. In other words, TX is the holomorphicvector bundle which is trivial over each Ui, and whose transition functionscorrespond to the Jacobian of the change of (holomorphic) coordinates fromthose defined by the chart φi to those defined by the chart φj .

Remark 1.36. Recall from the previous section the definition of the twosheaves of algebras AX and OX ; the stalks of these sheaves at a point x ∈ X,AX,x andOX,x, are the C-algebras of germs of differentiable and holomorphicfunctions respectively. A derivation of the C-algebra AX,x (resp. OX,x) is aC-linear map D : AX,x → C (resp. OX,x → C) that satisfies the Leibniz rule,i.e. for any f, g ∈ AX,x (resp. f, g ∈ OX,x), D(fg) = D(f)g(x) + f(x)D(g).The complexified tangent space TX,x,C is the space of derivations of AX,x,while the holomorphic tangent space TX is the space of derivations of OX,x.

Exercise 1.37. Let OCn,0 be the C-algebra of germs of holomorphic functionsat 0 ∈ Cn. Let z1, · · · , zn be standard coordinates on Cn and ∂

∂zi:= ∂

∂zi |0be defined by ∂

∂zi: f ∈ OCn,0 7→ ∂f

∂zi |0∈ C. Show that the ∂

∂ziare complex

derivations of OCn,0 and form a basis of TCn,0 over C. Deduce a basis ofTX,x for any complex manifold X.

Example 1.38. The tautological vector bundle U(r, V )→ Gr(r, V ) over theGrassmannian is defined as the vector bundle with total space

Ur(V ) = ([U ], x) ∈ Gr(r, V )× V |x ∈ U ⊂ Gr(r, V )× V,

and π : Ur(V ) → Gr(r, V ) the projection onto the first factor. Using thecomplex atlas of Gr(r, V ) determined in Example Sheet 1, show that Ur(V )is holomorphic.

Examples As in the differentiable situation, any canonical constructionin linear algebra gives rise to a geometric version for holomorphic vectorbundles. Assume that E and F are holomorphic vector bundles over acomplex manifold X, we can construct in this way:

E ⊕ F , the direct sum,

14

E ⊗ F , the tensor product,∧iE, the ith exterior product,

E∗, the dual bundle Hom(E,C) (i.e. fibre wise C-linear maps E → C),

detE the determinant line bundle.

In all cases, a common local trivialisation of E and F will yield transitionfunctions for these constructions. For example, if E is associated to thecocycle Ui, ei,j and F to the cocycle Ui, fi,j, E ⊕F is the vector bundle

of rank rkE + rkF associated to the cocycle Ui, gi,j =

(ei,j 00 fi,j

); E∗

is the vector bundle of rank rkE associated to the cocycle Ui, gi,j =t ei,j(Check that these are cocycles...).

Definition 1.39. Let X be a complex manifold and TX its holomorphictangent bundle. The cotangent bundle of X is the dual of the tangentbundle ΩX = T ∗X , the bundle of holomorphic k-forms is Ωp

X =∧p ΩX , the

canonical bundle is KX = det ΩX , and its dual is the anticanonical bundleK∗X = detTX .

Remark 1.40. The definitions of TX ,ΩX and KX are independent of thechoice of complex structure on X, they are invariant of the manifold X.

Definition 1.41. Let E1π1→ X and E2

π2→ X be two complex (resp. holo-morphic) vector bundles of rank r over C. Then E1 and E2 are isomorphicif there is a diffeomorphism (resp. biholomorphism) φ : E1 → E2 such thatπ1 = π2 φ.

Let X be a complex manifold and Ui, gi,j : Ui Uj → C∗ be a collectionof holomorphic functions that satisfy the cocycle condition. This defines aholomorphic line bundle L → X. The holomorphic line bundle L → Xis trivial (i.e. isomorphic to X × C) precisely when, possibly after refiningthe cover Uii∈I , there exist holomorphic functions si : Ui → C∗ such thatgi,j = si

sjon Ui ∩ Uj .

Definition 1.42. Let f : Y → X be a holomorphic map between complexmanifolds and E a holomorphic vector bundle over X. If Ui, gi,j is acocycle for E, define f∗E as the holomorphic vector bundle over Y associatedto the cocycle f−1(Ui), gi,j. For all y ∈ Y , there is a canonical isomorphism(f∗E)y ' Ef(y).

In particular, if Y is a submanifold of X, the restriction of E to Y isE|Y = i∗E, where i is the inclusion map i : Y → X.

15

Definition 1.43. Let π : E → X be a complex (resp. holomorphic) vectorbundle over a differentiable (resp. complex) manifold X. A global section ofE is a differentiable (resp holomorphic) map s : X → E such that π s = Id.

Remark 1.44. While a complex vector bundle always has many sections(because we may use local bump functions), this is not necessarily the casefor a holomorphic vector bundle.

Using fibrewise addition, the set of sections C∞(X,E) (resp. Γ(X,OX(E)))of a complex (resp. holomorphic) vector bundle has a natural C-vector spacestructure. If Ui, gi,j is a cocycle for E, over an open set Ui, C

∞(Ui, E)(resp. Γ(Ui, E)) has dimension rank r.

A basis si1 , · · · , sir of C∞(Ui, E) (resp Γ(Ui, E)) is a local frame (resp. lo-cal holomorphic frame) of E. The data of an open set Ui ⊂ X and a localframe si1 , · · · , sir is equivalent to a local trivialisation of E. If C∞(X,E),Γ(X,E) have dimension r, then a basis s1, · · · , sr is a global (holomorphicframe) of E, and E is trivial, i.e. isomorphic to X × Cr.

Example 1.45. Exercise 1.37 shows that for any complex chart U,ϕ ofa complex manifold X, if z1, · · · , zn are local holomorphic coordinates atu ∈ U , ∂

∂z1 |u, · · · ,∂∂zn |u is a local holomorphic frame of the holomorphic

tangent bundle TX .

Example 1.46. Recall the identification CPn = Pn = Gr(1, n + 1) anddefine the tautological line bundle OPn(−1) ⊂ Pn × Cn+1 → Pn as thetautological line bundle defined on Gr(1, n + 1) in Example 1.38. DefineOPn(1) = OPn(−1)∗ and OPn(k) = OPn(1)⊗k for k ∈ Z∗. Set OPn(0) for thetrivial line bundle.

If Ui = [l0: · · · :ln] ∈ Pn|li 6= 0, define si : Ui → OPn(−1) by:

[l0: · · · :ln] 7→ ([l]; (l0li, · · · , ln

li)).

The section si does not vanish anywhere on Ui; the associated local trivial-isation is:

hi : π−1(Ui)→ Ui × C([l], x) 7→ ([l], xi)

where xi is the unique complex number such that x = xisi([l]). The transi-

tion functions gi,j : (Ui ∩Uj)→ (Ui ∩Uj)×C∗ are gi,j : [l] 7→ ([l],ljli

) (this iswell defined on Ui ∩Uj). The sections of OPn(k) for k ∈ N are homogeneous

16

polynomials of degree k in the variables l0, · · · , ln. The transition functionsgi,j associated to the line bundle OPn(k) are of the form gki,j = si

sjfor si, sj

homogeneous polynomials of degree k.

Exercise 1.47. Describe the global sections Γ(Pn,OPn(k)) for k ∈ Z.

Definition 1.48. Let E → X be a complex (resp holomorphic) vectorbundle of rank R over a complex manifold X. A submanifold F ⊂ E is asubbundle of rank m if:

1. For all x ∈ X, F ∩ Ex is a subvector space of dimension m,

2. π|F : F → X has the structure of a complex (resp holomorphic) vectorbundle induced by that of E.

In other words, F is a subbundle of E if E and F are represented bycocycles Ui, ei,j and Ui, fi,j such that:

ei,j =

(fi,j ∗0 gi,j

)Examples

1. The tautological line bundle OPn(−1) is a subbundle of the trivialbundle Pn × Cn+1.

2. Let Y be a submanifold of X, TY is a subbundle of the restrictedtangent bundle TX |Y .

.

Definition 1.49. Let φ : E → F be a vector bundle homomorphism. Thereare well defined holomorphic vector bundles kerφ and Cokerφ.

If E and kerφ are represented by cocycles Ui, ei,j and Ui, ki,j with

ei,j =

(ki,j ∗0 gi,j

),

Cokerφ is associated to the cocycle Ui, gi,j.Sheaf theory was introduced as a unified way of dealing with problems

concerned with the passage from local data to global data; as such it is clearthat sheaves are useful to the study of (topological, differentiable, complex)manifolds and of (topological, complex, holomorphic) vector bundles overthese. Recall the definitions of sheaves:

17

Definition 1.50. Let X be a topological space. A presheaf F of abeliangroups (vector spaces, rings, algebras, etc..) on X consists of an abeliangroup (vector space, ring, algebra, etc..) Γ(U,F) = F(U) for every openset U ⊂ X, and a group homomorphism (linear map, ring homomorphism..)rU,V : F(U) → F(V ) associated to each pair of nested open sets V ⊂ U ,satisfying the compatibility conditions:

1. rU,U = Id for any U ⊂ X,

2. rU,W = rU,V rV,W for any W ⊂ V ⊂ U .

The presheaf F is a sheaf if it satisfies the further two conditions. DenoteU = ∪i∈IUi an open cover.

3. If s, t ∈ F(U) are such that rU,Ui(s) = rU,Ui(t) for all i ∈ I, then s = t,

4. If si ∈ F(Ui) is a collection of objects such that for all i, j ∈ I,rUi,Ui∩Uj (si) = rUj ,Ui∩Uj (sj), there is an element s ∈ F(U) such thatrU,Ui(s) = si for all i ∈ I.

The presheaves U 7→ C∞(U,E) and U 7→ Γ(U,E) are sheaves of abeliangroups on X, in both cases (complex and holomorphic vector bundles), wedenote E the sheaf of sections of E → X. If E is a complex (resp holomor-phic) vector bundle, E is a sheaf of AX -modules (resp OX -modules).

Remark 1.51. If R is a sheaf of rings over X, F is a sheaf of R-modulesover X if for every open U ⊂ X, F(U) has an R(U)-module structurecompatible with its group structure. The restriction maps F(U) → F(V )are morphisms of R(U) modules, where F(V ) is equipped with an R(U)-module structure via the restriction R(U)→ R(V ).

Definition 1.52. Let X be a connected topological space. A sheaf F ofR-modules over X is locally free of rank r if F is locally isomorphic to R⊕ras a sheaf of R-modules.

Proposition 1.53. Let X be a complex manifold. The map E → E is abijection between the set of holomorphic vector bundles E → X of rank rand locally free sheaves of rank r over X.

Proof. We have seen that if E → X is a holomorphic vector bundle, E is asheaf of OX -modules. For any local trivialisation, E|Ui ' Ui×Cr and hence

E|Ui ' O⊕rUi

and E is locally free of rank r. Conversely, let X = ∪Ui be

an open covering over which EUi ' O⊕rUi

is an isomorphism of sheaves. On

18

each Ui ∩ Uj , O⊕rUi∩Uj ' EUi∩Uj ' O⊕rUi∩Uj is an isomorphism of sheaves that

corresponds to an invertible r × r matrix MUi,Uj with holomorphic entries,i.e. to a holomorphic map gi,j : Ui ∩ Uj → GL(r,C). The sheaf axioms forE ensure that the maps gi,j are cocycles. Define E as the vector bundleassociated to the cocycle Ui, gi,j

Remark 1.54. Beware that when working with this bijection between thecategory of locally free sheaves and that of holomorphic vector bundles,morphisms of vector bundles are required to have constant rank, but mor-phisms of sheaves are not. This bijection in fact defines an equivalence ofcategories between the category of holomorphic vector bundles over X andthe category of free sheaves of OX -modules (this is one way to fix the rankof morphisms..).

Remark 1.55. An analogous statement holds for complex vector bundlesand locally free sheaves of AX -modules.

1.4 The complexified tangent bundle

We now start to examine systematically the relationship between the com-plex and differentiable structures on a complex manifold X of dimensionn.

Let V be an R-vector space of dimension m, an almost complex structureon V is an endomorphism I ∈ End(V ) such that I2 = − Id. An almostcomplex structure endows V with a structure of C-vector space via:

(a+ ib) · v = a · v + b · I(v), for v ∈ V, a and b ∈ R.

The dimension m has to be even, m = 2n. The C-vector space is denotedVC = V ⊗C. Define a complex conjugation on VC induced by I by: v ⊗ α =v ⊗ α, where v ∈ V and α ∈ C.

Conversely, a C-vector space W of dimension n is an R-vector spaceof dimension 2n, endowed with an almost complex structure I given bymultiplication by i.

If z1, · · · , zn are coordinates on Cn, denote xi = <zi and yi = =zi theassociated coordinates of R2n. The standard complex structure on Cn is theendomorphism of R2n given by the matrix whose only nonzero entries are n

blocks

(0 1−1 0

)on the diagonal.

Let (V, I) be an almost complex structure on a R-vector space V ofdimension 2n; I extends to an endomorphism of VC by I(v ⊗ λ) = I(v) ⊗

19

λ, and still satisfies J2 = − Id. The C-vector space endomorpshism I isdiagonalisable, and this gives a direct sum decomposition:

VC = V 1,0 ⊕ V 0,1,

where V 1,0 is the eigenspace associated to the eigenvalue i, and V 1,0 to −i.Define a conjugation on VC by v ⊗ α = v ⊗ α, we have V 0,1 = V 1,0.

Definition 1.56. Let X be a differentiable manifold of (real) dimensionm. An almost complex structure on X is a differentiable vector bundleisomorphism J : TX,R → TX,R such that J2 = − Id.

Remark 1.57. In general, there does not exist an almost complex structureon a differentiable manifold, even when its dimension is even. In fact, anyeven dimensional vector space admits a linear complex structure. As a con-sequence, for an even dimensional differentiable manifold X, we may alwaysdefine a linear transformation Ip : TX,P,R → TX,P,R such that I2

P = − Id. Theexistence of an almost complex structure on X is equivalent to determiningwhether this local construction can be patched up to a vector bundle diffeo-morphism (it is then uniquely determined by its action on each fibre). Thisbecomes a question of reduction of the structure group of the tangent bundlefrom GL2n(R) to GLn(C) and is a purely algebraic topological question.Thesphere S4 is an example of a differentiable manifold which admits no almostcomplex structure.

Proposition 1.58. Let X be a complex manifold, then X induces an almostcomplex structure on its underlying differentiable manifold.

Proof. If Ui, φi : Ui → Cn is a complex atlas of X, define a differentiableatlas on the underlying real manifold by Ui, (ui, vi) : Ui → R2n, whereui = <φi and vi = =φi. The real tangent bundle TX,R is trivial over Ui, and ∂∂ui1

, ∂∂vi1

, · · · , ∂∂uin

, ∂∂vin is a local frame. The holomorphic tangent bundle

TX is also trivial over Ui and a local holomorphic frame is ∂∂φi1

, · · · , ∂∂φin.

The identification

TX,R|U ' U × R2n ' U × Cn ' TX |U

endows TX,R with a (complex) vector bundle diffeomorphism I such thatI2 = − Id (I is induced by the standard complex structure on Cn). Thisdefinition is independent of the choice of chart in the complex atlas. Indeed,the transition functions φi,j = φi φ−1

j are holomorphic, so that the realJacobian of ψi,j , the corresponding differentiable transition functions ψi,j ,

20

JR(ψi,j) commutes with the matrix of I, the standard complex structureinduced by Cn. Recall that TX,R is given by Ui, JR(ψi,j ψ−1

j ) ψj, andby the Cauchy Riemann equations, JR(ψi,j) ψj is an n× n matrix of 2× 2

blocks of the form

(a b−b a

).

In fact, the same argument shows:

Lemma 1.59. Let f : U ⊂ Cn → V ⊂ Cm be a holomorphic map and x ∈ U ;then df(T 0,1

X,x) ⊂ T 0,1f(x) and df(T 1,0

X,x) ⊂ T 1,0f(x).

Let f : U ⊂ Cn → V ⊂ Cm be a holomorphic map; we also denote by fthe induced differentiable map R2n → R2m. The real Jacobian of f is:

JR(f) =

(∂uj∂xi

∂uj∂yi

∂vj∂xi

∂vj∂yi

)where the holomorphic coordinates on U are zi = xi + iyi, with xi, yi ∈ Rand fj = uj + ivj , where uj , vj : U → R. Recall that JR(f) viewed as amatrix with coefficients in C defines the transition functions of TX,R ⊗ C.When f is holomorphic, after an appropriate change of basis of TxR2n ⊗ Cand of Tf(x)R2m ⊗ C

JR(f) =

(J(f) 0

0 J(f)

),

where J(f) is the complex Jacobian of f .

Remark 1.60. This shows that every complex manifold has a natural ori-entation.

Proposition 1.61. Let X be a complex manifold. The subbundle T 0,1X ⊂

TX,R is naturally diffeomorphic to the holomorphic tangent bundle.

Proof. Let X be a complex manifold and Ui, φi : Ui → Cn a complexatlas. The complexified tangent bundle TX,C is represented by the cocycleUi, JR(φi φ−1

j )φj. When φi φ−1j is holomorphic, by what precedes, the

subbundle T 0,1X is associated to the cocycle Ui, J(φi φ−1

j ) φj. It follows

that T 0,1X and TX are naturally isomorphic as complex vector bundles.

Remark 1.62. When X is a complex manifold, this shows that T 0,1X natu-

rally has a holomorphic structure; however, we view it as a complex vectorbundle. In other words, sections of T 1,0

X are always differentiable sections.

21

We now go back to the general case; let (X, I) be an almost complexmanifold. There is a direct sum decomposition

TX,C = T 1,0X ⊕ T 0,1

X ,

where the summands are ker(I − i Id) and ker(I + i Id). This induces a dualdirect sum decomposition on the (complexified) cotangent complex:

ΩX,C = Ω1,0X ⊕ Ω0,1

X ,

and extends to a direct sum decomposition

ΩkX,C =

⊕p+q=k

Ωp,qX ,

where Ωp,qX =

∧p Ω1,0X ⊗C

∧q Ω0,1X . Further, Ω1,0

X = Ω0,1X . These decomposi-

tions are obvious at the vector space level (i.e. over a point x ∈ X), andthey are defined on the bundle via local trivialisations.

Definition 1.63. The sheaves AkX and Ap,qX are the sheaves of sections

of the complex vector bundles ΩkX,C and Ωp,q

X . Consider d : AkX → Ak+1X

the C-linear extension of the exterior differential. If Πk : A∗X → AkX andΠp,q : AkX → A

p,qX are the natural projection maps, define the operators ∂

and ∂ as ∂ = Πp+1,q d : Ap,q → Ap+1,q, and ∂ = Πp,q+1 d : Ap,q → Ap,q+1.

Lemma 1.64. The Leibniz rule holds for ∂ and ∂, i.e. if for some open setU ⊂ X, α ∈ AkX(U) and β ∈ Ak′X(U),

∂(α ∧ β) = ∂α ∧ β + (−1)kα ∧ ∂β∂(α ∧ β) = ∂α ∧ β + (−1)kα ∧ ∂β.

Proof. Follows from the Leibniz rule for d.

Definition-Lemma 1.65. Let (X, I) be an almost complex manifold, I isintegrable if the following equivalent conditions hold:

1. d = ∂ + ∂,

2. Π0,2 d = 0 on A0,1X .

If X is a complex manifold, the almost complex structure induced by I isintegrable.

22

Proof. We check that the conditions are indeed equivalent. If d = ∂ + ∂,since d2 = 0, Π0,2 d = 0. Conversely, assume that Π0,2 d = 0 on A1.0

X andlet α ∈ Ap,qX . We want to prove that dα ∈ Ap+1,q ⊕ Ap,q+1. Fix a common

trivialising subset U ⊂ X for Ω1,0X |U and Ω0,1

X |U and associated local frames

(ωi)1≤i≤n and (ω′i)1≤i≤n. Note that for any i, ω′i ∈ A1,0X (U). A section

α ∈ Ap,qX (U) is of the form

α = ΣI,JfI,JωI ∧ ωJ ,

where |I| = p and |J | = q, and where for each I, J , fI,J is a differentiablefunction on U, ωI = ωi1∧· · ·∧ωip , for I = i1, · · · , ip and ω′J = ω′j1∧· · ·∧ω

′jp

,

for J = j1, · · · , jp. For each I, J , dfI,J ∈ A1,0X (U) ⊕ A0,1

X (U) and dωi ∈A1,1X (U)⊕A2,0

X (U) by assumption, so that dfI,J ∧ ωI ∧ ωJ , and fIJdωI ∧ ωJare sections of Ap+1,q

X (U) ⊕Ap,q+1X (U). For any i, dω′i = dω′i, so that dω′i ∈

A2,0X (U)⊕A1,1

X (U) = A0,2X (U) ⊕ A1,1

X (U) and the result follows. When Xis a complex manifold, using trivialisations associated to any holomorphicchart, one sees that I is integrable.

Lemma 1.66. Let (X, I) be an integrable almost complex manifold. The

operators ∂ and ∂ satisfy ∂2 = ∂2

= ∂∂ + ∂∂ = 0.

Proof. This follows from d = ∂ + ∂ and d2 = 0.

Remark 1.67. In fact, one can show that if ∂2

= 0, I is integrable.

The notion of integrability is very important because of the following(hard) theorem, for which a proof in the real analytic case is given in [Voi02].

Theorem 1.68 (Newlander-Nirenberg Theorem). Any integrable almostcomplex structure is induced by a complex structure.

Remark 1.69. It would be natural to ask how much of complex geometricmethods can be applied to the setting of non-integrable almost complexmanifolds. The proof of the Newlander-Nirenberg theorem relies on findingholomorphic coordinates near each point. If (X, I) is an almost complexmanifold, a differentiable function f : X → C is I-holomorphic if Idf = idf ,that is if du = Idv and dv = −Idu, where u = <f and v = =f . WhenX has dimension 2n, this is a set of 2 equations in 2n variables, and theremay be few holomorphic functions. When I is integrable, the derivativesof holomorphic functions span (T 1,0

X,R,x)∗ for all x ∈ X, and this yields acomplex structure on X. When dimX = 2, an almost complex structure

23

is always integrable. On a general almost complex manifold however, thereare in general few holomorphic functions f : X → C. In particular, thereare few submanifolds Y ⊂ X defined by holomorphic injections of dimension2 ≤ dimY ≤ dimX. The complex submanifolds Y ⊂ X of dimension 2 areI-holomorphic curves, and are in general well behaved, they are studied inSymplectic Geometry.

Proposition 1.70. Let X,Y be complex manifolds and f : X → Y a holo-morphic map. The pull back of differential forms respects the decompositionof k-forms into forms of type (p, q), i.e. f∗Ap,qY ⊂ A

p,qX for all p, q, and f∗ is

C-linear and compatible with ∂ and ∂.

Proof. Since f is differentiable, the pull back map f∗ : AkY → AkX satisfiesf∗ dY = dX f∗. The pullback is C-linear, and as in Lemma 1.59, f∗

respects the decomposition of k-forms into forms of type (p, q) for p+ q = kbecause f is holomorphic. The compatibility with ∂ and ∂ follows from thecompatibility with d.

1.5 Sheaf cohomology and Dolbeault Cohomology groups

Let X be a complex manifold, we have seen that for integers p, q we maydefine vector bundles

Ωp,qX =

p∧Ω1,0X ⊗

q∧Ω0,1X .

Let Ap,qX denote the sheaf of (C∞) sections of the bundle Ωp,qX . The exterior

diffential d = ∂ + ∂ induces differential operators

Ap,q ∂→ Ap+1,q and Ap,q ∂→ Ap,q+1

with ∂2 = ∂2

= 0. These operators define cohomological complexes, andwe now examine the cohomology of the Dolbeault complex, associated to∂. Denote Zp,q

∂(U) = α ∈ Ap,qX (U) : ∂α = 0 and Bp,q

∂(U) = ∂β;β ∈

Ap,q−1(U). The Dolbeault cohomology groups of X are:

Hp,q

∂(X) = Zp,q

∂(X)/Bp,q

∂(X).

We first examine the case q = 0.

Lemma 1.71. For all p ≥ 0:

Hp,0

∂(X) = Γ(X,Ωp

X) := H0(X,ΩpX).

24

Proof. First note that since q = 0, Hp,0

∂(X) = Zp,0

∂(X) = ker(∂ : Ap,0(X)→

Ap,1(X)). Locally, if z1, · · · , zn is a set of holomorphic coordinates on X,dz1, · · · , dzn is a local frame for Ω1,0

X and dz1, · · · , dzn is a frame for Ω0,1X .

The forms dzI ∧ dzJ where I, J run through multiindices with |I| = p and|J | = q form a frame for Ωp,q

X . If α ∈ Ap,0(X), locally, α = ΣIαIdzI , whereαI is a differentiable function, and:

∂α = Σnj=1ΣI

∂αI∂zj

dzj ∧ dzI .

The dzj∧dzII,j form a local frame of Ωp,1X , so that if ∂α = 0, ∂αI∂zj

= 0 for all

I, j. Each αI is a holomorphic function, and by the canonical identificationbetween the holomorphic cotangent bundle ΩX and Ω1,0

X , this shows that αis a holomorphic section of Ωp

X .

We will extend this statement to the higher cohomology groups of theholomorphic vector bundle Ωp

X .

Proposition 1.72. The Dolbeault complex Ap,•, ∂ is an exact complex ofsheaves and is a resolution of Ωp

X , i.e. the sequence of sheaves

0→ ΩpX → A

p,0X

∂→ Ap,1X∂→ · · · ∂→ Ap,nX → 0

is exact.

Let X be a topological space and E ,F and G be sheaves of abeliangroups/ modules, etc.. over X (see definition 1.50). A morphism of sheavesα : F → G is a collection of homomorphisms α(U) : F(U) → G(U) for eachopen set U ⊂ X that are compatible with the restriction maps rF and rG ,i.e. for every pair of open sets V ⊂ U ⊂ X, rGUV α(U) = α(V ) rFUV . Ifα is a morphism of sheaves, kerα is the sheaf U 7→ kerα(U) and Cokerαand imα are the sheaves associated to the presheaves U 7→ Cokerα(U) andU 7→ imα(U) respectively.

Remark 1.73. These two last presheaves are not sheaves in general.. E.g.,if exp(2iπ·) : OX → O∗X , the map z 7→ z ∈ O∗U is not in α(OU ) forU = Cr 0, but z 7→ z ∈ α(OV ) for any contractible V ⊂ U .

Given a morphism of sheaves α : F → G, a section s ∈ Cokerα is anopen cover X = ∪i∈IUi and a collection of sections si ∈ G(Ui) such thatsi|Ui∩Uj − sj |Ui∩Uj ∈ α(Ui ∩ Uj)(F(Ui ∩ Uj)). Two such sections Ui, sii∈I

25

and Vj , tj are identified if for any P ∈ Ui ∩ Vj , there is a neighbourhoodV of P such that si|V − tj |V ∈ α(V )(F(V )).

A sequence of sheaf morphisms 0→ E α→ F β→ G → 0 is exact if E = kerαand G = Cokerβ. A sequence of sheaves

· · · αn−1→ Fnαn→ Fn+1

αn+1→ · · ·

is exact if for each n ∈ N, kerαn+1 = imαn, i.e. if for each n ∈ N,

0→ kerαn → Fn → kerαn+1 → 0

is exact.

Remark 1.74. Let 0 → E α→ F β→ G → 0 be a short exact sequenceof sheaves. Let U ⊂ X be an open set, 0 → E(U) → F(U) → G(U) isexact (i.e. α(U) is injective, and imα(U) = kerβ(U)) but β(U) is not ingeneral surjective. As we will see, the failure of surjectivity is “measured”by the cohomology of the sheaves E ,F ,G. Note that the sequence of stalks

0→ EPα→ FP

β→ GP → 0 is exact for each P ∈ X.

Proof of 1.72. Lemma 1.71 shows that ΩpX = ker(∂ : Ap,0X → A

p,1X ) so that:

0→ ΩpX → A

p,0X → A

p,1X

is exact. We want to show that for each p,

im(∂ : Ap,q−1

X → Ap,qX)

= ker(∂ : Ap,qX → A

p,q+1X

).

This follows from Lemma 1.75.

Lemma 1.75 (∂-Poincare lemma). Let U ⊂ Cn be an open neighbourhoodof D(0, R), the closure of a bounded polydisc and let α ∈ Ap,qX (U) a ∂-closed

form. There is a polydisc D′ ⊂ D and β ∈ Ap,q−1X (D′) such that α|D′ = ∂β.

Proof. We first reduce to the case p = 0.Let α be a section of Ap,qX (U). Locally, we may write

α = ΣI,JαI,JdzI ∧ dzJ = ΣJαIdzI ,

where the first sum runs over multi-indices I and J of length p and q re-spectively, and where αI = ΣαI,JdzJ ∈ A0,q(U). Write:

∂α = Σnl=1ΣI,J

∂αIJ∂zl

dzl ∧ dzI ∧ dzJ = Σj,l∂αJ∂zl

dzl ∧ dzI .

26

Since dzl ∧ dzI ∧ dzJ is a local frame for Ωp,q+1X , ∂α = 0 precisely when

∂αJ = Σl∂αJ∂zl

dzl = 0 for all J . Similarly, using local frames, wwe see that

α = ∂β, precisely when αI = ∂βI for all I.We now assume that α ∈ A0,q(U) is such that ∂α = 0. Locally, write

α = Σ|J |=qfJdzJ and define:

k = minl : no dzi appears in α for i > l.

We may write α = α1 ∧ dzk + α2, where no dzi for i ≥ k appears in α1 ∈A0,q−1(U) or α2 ∈ A0,q(U). Define operators ∂i by:

∂ = Σni=1

∂

∂zidzi = Σn

i=1∂i.

If ∂α = 0, we immediately see that ∂i(α1) = ∂i(α2) = 0 for all i ≥ k + 1, sothat all fJ are holomorphic in the variables zk+1, · · · , zn. By the ∂-Poincarelemma in 1-variable (see Example Sheet 1), for some 0 < εk < Rk, if

gJ(z) =1

2iπ

∫D(0,εk)

fJ(z1, · · · , zk−1, ω, zk+1, · · · , zn)

ω − zkdω ∧ dω,

then, on D′ = D(0, R′) for R′ = (R1, · · · , Rk−1, εk, Rk+1, · · · , Rn), ∂gJ∂zk= fJ

and gJ is C∞ and holomorphic in the variables zk+1, · · · , zn. Set γ =(−1)qΣk∈JgJdzJ−k, then α+∂γ is ∂-closed and does not involve any mono-mial dzl for l ≥ k. We conclude by induction on k.

We have proved that the Dolbeault complex Ap,•, ∂ is an exact complexof sheaves, the proof of Lemma 1.71 shows that this complex is a resolution ofthe sheaf Ωp

X . We now show that the cohomology of the Dolbeault complexcomputes the cohomology of the sheaf Ωp

X .We first recall the setup of Cech cohomology. Let X be a (paracompact)

topological space, F a sheaf on X and U = Uii∈I a locally finite opencover of X. Define the Cech complex (C•(U ,F), δ) of F associated to thecover U as:

Cp(U ,F) = Π|J |=p+1F(UJp),

where J runs through multi-indices of length p+ 1, and UJ = Ui0 ∩ · · · ∩Uipfor J = i0, · · · , ip ⊂ I, and the boundary map

δ : Cp(U ,F)→ Cp+1(U ,F),

sends σ to the p+ 1-chain whose component on F(Ui0 ∩ · · ·Uip+1) is:

δ(σ)i0,··· ,ip+1 = Σp+1j=0(−1)j(σi0,··· ,ij ,··· ,ip+1

)|Ui0∩···∩Uip .

27

Note that this is a cohomological complex, i.e. δ2 = 0. An elements σ ∈Cp(U ,F) is a p-cochain, if δσ = 0, σ is a cocycle, and if σ = δτ for τ ∈Cp+1(U ,F) is a boundary. Denote Zp(U ,F) = σ ∈ Cp(U ,F)|δσ = 0 andBp(U ,F) = δτ ; τ ∈ Cp+1(U ,F); the Cech cohomology groups of U are:

Hp(U ,F) = Zp(U ,F)/Bp(U ,F).

Assume that U = Uii∈I and V = Vjj∈J are two covers; V is finerthan U if for any j ∈ J , there is i ∈ I such that Vj ⊂ Ui. If V is finer thanU , define a map ϕ : J → I by sending j ∈ J to an index ϕ(j) ∈ I such thatVj ⊂ Uϕ(j); the map ϕ induces a map ρϕ : Cp(U ,F) → Cp(V,F) such that

δVρϕ = ρϕδV and hence a homomorphism ρ : Hp(U ,F)→ Hp(V,F). Thishomomorphism is well defined because different choices ϕ,ψ : I → J inducechain homotopic maps ρϕ and ρψ. Define the Cech cohomology groups of Fas the direct limit of Hp(U ,F) as U gets finer:

Hp(X,F) = lim→Hp(U ,F).

This definition is difficult to work with on an arbitrary topological space X.However, note that, for any cover U of X:

H0(U ,F) = Γ(X,F) := H0(X,F).

It is useful to know which open covers actually compute Hp(X,F).

Theorem 1.76 (Leray theorem). If the cover U is acyclic for F ,i.e. for anyintersection of open sets UJ = Ui0 ∩ · · · ∩ Uip with p > 0, and for all q > 0:

Hq(UJ ,F) = (0),

thenH•(U ,F) = H•(X,F).

Proposition 1.77. If X is separated and if F is a quasi-coherent sheaf,any cover by affine subschemes is a Leray cover.

Remark 1.78. When X is a complex manifold and F is the sheaf of sectionsof a complex or holomorphic vector bundle, any cover by open affine sets isa Leray cover.

One way to think about Cech cohomology is as follows. If U is an opencover of X, then for any p, the (p + 2)-fold intersections of open sets in UUJ ; |J | = p+2 form an open cover of ∪|J ′|=p+1UJ ′ , the union of (p+1)-fold

28

intersections of open sets in U . A cochain σ ∈ Cp(U ,F) is a collection ofsections σJ ∈ F(UJ) for all |J | = p + 1. If σ is a cocycle, definition 1.50shows that σ defines a section of F(∪|J |=p+1UJ), if σ is a boundary, σ isthe restriction of a section of F(∪|J |=pUJ). If the cover U is acyclic, the

classes Hp(X,F) corespond to the sections of F(∪|J |=p+1UJ) that are notrestrictions of sections of F(∪|J |=pUJ).

The important fact about Cech cohomology–which shows that this def-inition of sheaf cohomology agrees with that of Grothendieck, using rightderived functors of the functor of global sections– is that, for any short exactsequence of sheaves

0→ E α→ F β→ G → 0

the sequence

0→ H0(X, E)→ H0(X,F)→ H0(X,G)→ H1(X, E)→ · · ·→ · · ·Hp−1(X,G)→ Hp(X, E)→ Hp(X,F)→ Hp(X,G)→ · · ·

is a long exact sequence of cohomology groups.Assume that U is an acyclic cover for E ,F and G. The definition of

cohomology groups is functorial ; if α : E → F is a sheaf homorphism, thereis a well defined map α∗ : Cp(U , E) → Cp(U ,F), which commutes with theboundary maps δ of the Cech complexes. In particular, α, β induce maps

Hp(U , E)α∗→ Hp(U ,F) and Hp(U ,F)

β∗→ Hp(U ,G)

and hence maps Hp(X, E)α∗→ Hp(X,F) and Hp(X,F)

β∗→ Hp(X,G).

For every p ∈ N, the coboundary map Hp(X,G)δ∗→ Hp+1(X, E) is defined

as follows. Let [σ] ∈ Hp(X,G), and take a representative σ ∈ Zp(U ,G) of[σ] for some cover U . Since G = imβ, there is a refinement V of U –because U 7→ imβ(U) is a presheaf and not a priori a sheaf– and a chainτ ∈ Cp(V,F) with βτ = ρVσ. The cochain δτ is a section of kerβ becauseβδτ = δβτ = δρσ = 0, and since kerβ = imα, there is a refinement W ofV and a cochain µ ∈ Cp+1(W, E) such that αµ = ρWδτ . The cochain µ isa cocycle because αδµ = δαµ = δρWδτ = ρWδ

2τ = 0 and α is injective, sothat δµ = 0. Define δ∗[σ] = [µ] ∈ Hp+1(X, E).

In this course, we will consider three types of sheaves, each type willcarry information of a specific nature on a complex manifold X.

• Holomorphic sheaves such as OX , TX or ΩpX carry much information

about the geometry, complex structure and deformations of the mani-fold. Studying the cohomology of holomorphic sheaves will emphasize

29

properties that are particular to the complex structure, rather thandepending only on the differentiable structure. Holomorphic sheaveshave few sections (see Example sheets 1,2), but by analytic continua-tion, when a section exists, knowing its restriction to a small open setdetermines it entirely.

• Differentiable sheaves such as ApX ,Ap,qX , TX,R, TX,C will mainly be aux-

iliary tools from the point of view of complex geometry. This idea isalready at work in the statements of Lemmata 1.71 and 1.75. On acomplex compact manifold, differentiable sheaves are useful becausethey have many sections. They are sheaves with “partitions of unity”,so it is straightforward to extend sections.

• Topological sheaves such as Z,R and C. The cohomology of thesesheaves exhibits topological properties of the underlying space. Amajor idea of Hodge theory is to relate the topology ofX to its complexstructure.

We now show that sheaves with partition of unity (such as Ap,q) areacyclic.

Let X be a paracompact complex manifold. Let F be a differentiablesheaf (a sheaf ofA0

X -modules). For any locally finite open cover U = Uii∈I ,there is a partition of unity, that is a collection of differentiable functionsρi : X → R such that Supp ρi ⊂ Ui for all i ∈ I and such that Σρi ≡ 1.Given σ ∈ Zp(U ,F), then σ = δτ , for τ ∈ Cp−1(U ,F) defined by τi0,···ip−1 =Σk∈Iρkσk,i0,··· ,ip−1 . We have just shown:

Lemma 1.79. If X is a complex manifold, any differentiable sheaf F (suchas Ap,qX ,AkX ..) is acyclic, i.e. for all i > 0:

H i(X,F) = (0).

Lemma 1.80. If 0 → F → F0 → F1 → · · · is the resolution of a sheaf Fby an exact complex F•, δ of acyclic sheaves, then Hp(XF) ' Hp

δ (X,F•),where

Hpδ (X,F•) = ker(δp : Fp(X)→ Fp+1(X))/ im(δp−1 : Fp−1(X)→ Fp(X)).

Proof. Denote Zp(δ)(X,F•) = ker(δp : Fp(X)→ Fp+1(X)) andBpδ (X,F•) =

im(δp−1 : Fp−1(X) → Fp(X)). If 0 → F → F0 → F1 → · · · is a resolutionof F , then by definition, the sequences

0→ F(X)→ F0(X)→ Z1δ (X) → 0

0→ Zpδ (X)→ Fp(X)→ Zp+1δ (X) → 0

30

for all p ∈ N. Since H i(X,Fp) = (0) for all i > 0 and p ≥ 0, we obtain:

Hq(X,F) ' Hq−1(X,Z1δ (X)) ' · · · ' H1(X,Zp−1

δ (X))

' H0(X,Zpδ (X))/δH0(X,Fp−1δ (X)) ' Hp

δ (X,F•).

Recall that Lemmata 1.71 and 1.75 show that the Dolbeault complexAp,•X , ∂ is a resolution of Ωp

X , and by Lemma 1.79, the sheaves Ap,qX areacyclic for all p, q. We have proved:

Theorem 1.81. For all p, q ∈ N:

Hq(X,ΩpX) ' Hp,q

∂(X).

Example 1.82. Let X = Cn, Hk(X,OX) = H0,k

∂(X) = 0 for all k ∈ N.

Example 1.83. Let E → X be a holomorphic vector bundle, and Ap,qX (E)the sheaf of differentiable sections of the vector bundle Ωp,q

X ⊗ E. Define

a sheaf homomorphism ∂E : Ap,qX (E) → Ap,q+1X (E) as follows. Let U be an

open set and E|U ' U×Cr be a local trivialisation of E. If α⊗s is a section

of Ap,qX (E)(U), ∂E(α⊗ s) = ∂(α)⊗ s. The kernel of ∂E : A0,0X (E)→ A0,1

X (E)is E , the sheaf of holomorphic sections of E (this is entirely similar to the

proof of Lemma 1.71). The operator ∂E satisfies the Leibniz rule and ∂2E = 0.

Local exactness of the complex Ap,•, ∂E follows from that of the Dolbeaultcomplex (Lemma 1.75). In particular,

0→ E → A0,0X (E)

∂E→ A0,1X (E)

∂E→ · · ·

is a resolution of the holomorphic sheaf of sections E . This resolution isacyclic, and Hp(X, E) ' H0,p

∂E(X).

We will now look at another application of sheaf cohomology: the DeRham theorem. We denote AkX the sheaf of real or complex k-forms; thiswill not lead to any (major) confusion. The ordinary Poincare lemma statesthat the (real) De Rham complex A•X , d is a resolution of the sheaf R oflocally constant functions:

0→ R→ A0X

d→ A1X → · · ·

By Lemma 1.79, the De Rham complex is an acyclic complex of sheaves, andtherefore the Betti cohomology of X, that is the cohomology of the sheaf R

Hk(X,R) ' HkDR(X).

31

In fact, more is true. If X is a differentiable manifold, X = |K| is thegeometric realisation of a simplicial complex K. Recall that the singularcohomology of X is then

Hksing(X,R) = Hk

sing(X,Z)⊗ R = Hom(Hsingk (X,Z),R),

where Hsingk (X,Z) the homology of the complex Csing• (X,Z), ∂ of singular

chains. Recall from Algebraic Topology that Csingp (X,Z) is the free abeliangroup generated by continuous maps ϕ : ∆p → X, where ∆p is the stan-dard p-simplex. The boundary map is ∂ϕ = Σ(−1)iϕ|∆i

k, where ∆i

k is theith face of ∆k. If X is the geometric realisation of an abstract simplexK, Hk

sing(X,Z) ' Hk(X,Z). If K is a simplicial complex with verticesvio , · · · , vin, we can associate to K an open cover U of X as follows. Foreach vertex vi ∈ K, define Ui as the Star of vi, i.e. the union of the interiorsof all simplices of which vi is a vertex. Then vi0 , · · · , vip span a p-simplexprecisely when Ui0 ∩ · · · ∩ Uip 6= ∅. A Cech cochain σ ∈ Cp(U ,Z) is then acollection of sections of Z(Ui0,··· ,ip) (this is Z if vi0 , · · · , vip span a p-simplex,and 0 otherwise). To σ, we associate the cochain σ′ ∈ Cpsing(X,Z), the mapσ′ : ∆ =< vi0 , · · · , vip >7→ σi0,··· ,ip . This correspondence σ → σ′ is an iso-morphism of abelian groups, and since ∂σ′ = (δσ)′, we have an isomorphismof chain complexes C•(U ,F)→ Csing• (X,Z), and hence:

Hk(U , Z) ' Hksing(X,Z).

Note that refining the cover U corresponds to subdividing the simplicialcomplex K, and since this does not change the singular cohomology, U is aLeray cover. We have proved:

Theorem 1.84. For a differentiable manifold X and for all k ∈ N:

Hk(X,Z) ' Hksing(X,Z) and Hk

DR(X) ' Hk(X,R) ' Hksing(X,R).

Remark 1.85. The sheaf R may be replaced by C-provided the De Rhamcomplex is taken to be the complex of C-valued forms. As I have mentionedabove, in the cases we consider, Cech cohomology always agrees with sheafcohomology. I have already written H instead of H several times. Fromnow on, I always write H.

A more detailed version of Theorem 1.84 is known as the De RhamTheorem. Results in Differentiable Topology imply that the cohomology ofC•sing(X,R), ∂ coincides with that of the subcomplex of piecewise smooth

cochains (Cp.s.)•sing(X,R), ∂. Let [γ] ∈ Hksing(X,Z) be represented by

32

γ = Σaifi, where fi : ∆→ R is piecewise smooth–i.e. extends to a C∞ mapin a neighbourhood of ∆– and [α] ∈ Hk

DR(X) be represented by a k-formα ∈ AkX(X). Define a pairing:

< α, γ >=

∫γα = Σai

∫∆f∗i α

with values in R. The Stokes formula implies that this pairing asociates to[α] a cohomology class in Hk

sing(X,R). Indeed, for any β ∈ Ak−1(X),∫γα =

∫γα+ dβ,

and if dα = 0, ∫∂εα =

∫εdα = 0

for any boundary of a k + 1-chain ε.

Theorem 1.86 (De Rham Theorem). Let X be a differentiable manifold.The map

α 7→ (γ →∫γα)

is an isomorphism and provides a canonical identification HkDR(X,R) '

Hksing(X,Z)⊗ R.

2 Hermitiann and Kahler metrics on complex man-ifolds

2.1 Hermitian metrics and connections of vector bundles

The concepts of Hermitian metrics, connections and curvature for complexvector bundles are very close to the notions of Riemannian metrics on realdifferentiable manifolds.

Definition 2.1. Let π : E → X be a complex vector bundle over a dif-ferentiable manifold. Let AkX(E) be the sheaf of differentiable sections ofthe vector bundle Ωk

X,C ⊗ E. A connection on E is a C-linear sheaf homo-

morphism D : A0X(E) → A1

X(E) that satisfies the Leibniz rule D(f ⊗ s) =df ⊗ s+ f ⊗Ds.

33

Remark 2.2. A connection induces sheaf homomorphisms D : AkX(E) →Ak+1X (E), with D(τ ⊗ s) = dτ ⊗ s + (−1)kτ ⊗ Ds for τ ∈ AkX(U) and

s ∈ A0X(E)(U) over a trivisalising open set U for the vector bundle E.

Example 2.3. Let E ' X × Cr be the trivial vector bundle; define D(α⊗s) = dα ⊗ s for local sections α ∈ AkX(U) and s ∈ A0

X(U). The operatorD is the trivial connection: it acts as the exterior differential d on the formpart of sections of AkX(E).

If E → X is a holomorphic vector bundle over a complex manifold, theoperator ∂E in Example 1.83 is a connection.

If D and D′ define two connections on E, D − D′ is A0X -linear, that

is (D − D′)(fs) = f(D − D′)(s). The operator D − D′ hence is a globalsection of the sheaf A1

X(EndE). We write Ak(X,F ) = C∞(X,ΩkX,C ⊗ F )

for global sections of AkX(F ), where F is a complex vector bundle. Leta ∈ A1(X,EndE), a acts on a local section α⊗s of Ak(E) by multiplicationon the form part and evaluation End(E)×E → E on the bundle component.

If E is a trivial vector bundle, let d be the trivial connection definedin Example 2.3. Any connection D on E is of the form d + a, where a ∈A1(X,EndE) = A1

X ⊗ EndE.We want to deduce from this the form of connections on an arbitrary

vector bundle E → X. Let Ui, hi : E|Ui ' Ui × Cr be local trivialisationsof E, and recall that we denote gi,j the induced transition map Ui ∩ Uj →GLr(C). In a local frame, sections of EndE are represented by (r × r)matrices with differentiable entries, and sections of A1

X(EndE) by (r × r)-matrices of 1-forms. If d is the trivial connection on E|Ui , the restrictionDi = D|Ui : A0(E)(Ui) → A1(E)(Ui) is of the form di + Ai, where Ai is a(r × r)-matrix of 1-forms. On Ui ∩ Uj , we have the transition relations:

Dj = h−1j (dj +Aj) hj , and Aj = g−1

ij dgij + g−1ij A

igij . (2)

Definition-Lemma 2.4. The curvature of D is

ΘD = D2 : A0(E)→ A2(E).

The curvature ΘD is A0-linear, i.e. defines a global section of A2(EndE).

Proof. If f is a differentiable function, D2(fs) = fD2(s).

Let D be a connection on E and a ∈ A1(X,EndE), then

ΘD+a = ΘD +D(a) + a ∧ a,

where a ∧ a ∈ A2(X,EndE) acts by exterior product on the form part andby composition in EndE.

34

Definition 2.5. A Hermitian metric h on E → X is a Hermitian innerproduct hx : Ex × Ex → C on each fibre Ex such that for any open setU ⊂ X and s1, s2 ∈ A0(E)(U) local sections of E over U ,

h(s1, s2) : U → Cx→ hx(s1(x), s2(x))

is differentiable. A Hermitian vector bundle (E, h)→ X is a complex vectorbundle endowed with a Hermitian metric.

Proposition 2.6. Every complex vector bundle E → X is Hermitian.

Proof. Let Ui, hi : E|Ui ' Ui×Cri∈I be local trivialisations of E → X. Letei1, · · · , eir the local frame associated to hi. In that local frame, a Hermitianmetric h is represented by an (r× r)-matrix of differentiable functions H i =(hk,l)1≤k,l≤r, where hk,l(x) = hx(ek(x), el(x)). If s1, s2 ∈ A0(E)(Ui) arelocal sections of E, s1 = Σsi1,ke

ik and s2 = Σsi2,ke

ik, then h(s1, s2)|Ui =

Σsi1,khi(k, l)si2,l. Note that if x ∈ Ui ∩ Uj , Hj = gti,jH

igij .Define a Hermitian metric h on E → X by setting

hi(si1, si2) = Σr

k=1si1,ks

i2,k

and h = Σρihi, where ρii∈I is a partition of unity subordinate to the open

cover Uii∈I of X.

Note that applying the Gram-Schmid orthonormalisation process, wemay always choose local frames of E → X in which the Hermitian metricon each fibre Ex is the standard Hermitian metric of Cr. Such frames arecalled isometric.

Definition 2.7. Let (E, h) → X be a Hermitian vector bundle and D aconnection on E. The connection D is compatible with the metric h if forany two local sections s1, s2 ∈ A0(E)(U) over an open set U ⊂ X,

dh(s1, s2) = h(Ds1, s2) + h(s2, Ds1),

where for any two 1-forms α, β ∈ A1X(U), h(α ⊗ s1, s2) = αh(s1, s2) and

h(s1, β ⊗ s2) = βh(s1, s2).

Assume that in a local trivialisation E|U ' U ×Cr, D = d+A, where Ais an (r × r)-matrix of 1-forms. Then, D is Hermitian if:

dH = At ·H +HA. (3)

35

If the local frame is isometric, D is Hermitian precisely when At

= −A. Wesay that D is compatible with the metric.

If D is a connection given by d+ A in a local trivialisation of E, definethe adjoint connection Dadj as the connection given by the matrix

Aadj = −At

(check that this defines a connection: i.e. satisfies the transition formulae of(2)). The connection Dh = 1

2(D +Dadj) is always a Hermitian connection.

Now assume that E → X is a holomorphic vector bundle over a complexmanifold. We have seen that there is a direct sum decomposition

A1(E) = A1,0(E)⊕A0,1(E),

and this induces a decomposition D = D1,0 +D0,1. Consider a local trivial-isation of E, and let f ∈ A0(U) and s ∈ A0(E)(U), then:

D1,0(fs) = ∂f · s+ f ·A1,0s and D0,1(fs) = ∂f · s+ f ·A0,1s,

where A1,0 (resp. A0,1) is an (r×r)-matrix of (1, 0)-forms (resp. (0, 1)-forms).

Definition 2.8. The connection D is compatible with the holomorphicstructure if D0,1(fs) = ∂f · s, i.e. if D0,1 = ∂E is the Dolbeault connec-tion of Example 1.83.

Proposition 2.9. Let (E, h)→ X be a holomorphic Hermitian vector bun-dle. There is a unique Hermitian connection DE that is compatible with theholomorphic structure.This connection is the Chern connection of E → X;its curvature ΘE is the Chern curvature.

Proof. We first prove that if such a connection exists, it is uniquely de-termined. A connection is a sheaf homomorphism, hence uniqueness is alocal question. Let E|U ' U × Cr be a local trivialisation and assumethat in the associated local frame, the connection D is given by d + A,where A = A1,0 + A0,1 is an (r × r)-matrix of 1-forms, and h is given byan (r × r)-Hermitian matrix of differentiable functions. Since D is Hermi-tian, dH = AtH +HA. If D is compatible with the holomorphic structure,A = A1,0 and ∂H = AtH and ∂H = HA. In particular,

A = H−1∂H, (4)

and D is uniquely determined by H.

36

In order to prove the existence of the connection DE , it is enough toprove that (4) defines a connection. We want to prove that if Ui, hi : E|Ui 'Ui × Cr is a system of local trivialisations for E, the Di

E defined by (4)define a connection globally, i.e. satisfy the transition relations (2). This isleft as an exercise.

Example 2.10. 4.1 Let L→ X be a holomorphic line bundle. A Hermitianmetric on L is a positive real valued function h : X → R∗+ given by h(x) =|e1(x)|2, where e1 is a local holomorphic frame of L. The Chern connectionof (L, h) is then DE = d+ log h.

Corollary 2.11. Let (E, h) → X be a holomorphic Hermitian vector bun-dle over a complex manifold X and DE ,ΘE be its Chern connection andcurvature. If, in a local holomorphic frame, DE = d+A, then

1. A is of type (1, 0) and ∂A = A ∧A,

2. ΘE = ∂A is of type (1, 1), so that ΘE ∈ A1,1(X,EndE),

3. ∂ΘE = 0, the 2-form ΘE is ∂-closed.

Proof. This all follows easily from the relation (4) and from the local formof the curvature ΘE = dA+A ∧A.

Remark 2.12. We have seen that if (E, h)→ X is a Hermitian (complex)vector bundle over a differentiable manifold, we may always pick isometriclocal (differentiable) frames for E. In such a frame, the Hermitian metriccoincides with the standard Hermitian structure on Cn, and its matrix is theidentity. The Gram-Schmid orthonormalisation process is not holomorphic,so in general we cannot hope to find isometric holomorphic local frames.

In any local holomorphic frame, the Hermitian metric is represented bya hermitian matrix H(z) with differentiable entries. However, if P ∈ X is apoint and E|U ' U × Cr is a local holomorphic frame in a neighbourhoodof P , and z1, · · · , zn are local holomorphic coordinates centred at P , H(0)is a positive definite Hermitian matrix, so that there is B ∈ GL(Cr) withBtH(0)B = Id. The matrix B determines a linear (hence holomorphic)change of coordinates for the local frame of E over U such that H(0) = Id–that is such that the restriction of the Hermitian metric to the fibre overP ∈ X is the standard Hermitian metric on Cr.

Lemma 2.13. Let (E, h) : X be a holomorphic, Hermitian vector bundleover a complex manifold X, and P ∈ X a point. There is a local holomorphicframe of E near P inducing local holomorphic coordinates z1, · · · , zn onP ∈ U ⊂ X such that on U :

37

1. H(z) = Id +O(|z|2), and

2. iΘE(0) = −i∂∂H(0).

Proof. The second assertion follows from the first by Corollary 2.11 and the

relation A = H−1∂H. Take a local holomorphic frame for E such that in

that frame H(0) = Id. We now define a holomorphic change of coordinatesonX given by a matrix Id +B, where B is a matrix with holomorphic entries,such that the first assertion holds. After such a change of coordinates, thematrix of the Hermitian metric in the new frame is

H ′(z) = (Id +B(z)t) ·H(z) · (Id +B(z).

Denote B = (Bj,k)1 ≤ j, k ≤ r, and set

Bj,k(z) = Σni=1

∂Hk,j

∂zi(0)zi,

so that dH ′(0) = (0), and we conclude by the Taylor formula.

2.2 Kahler metrics

Let VC be an n-dimensional C-vector space, write VR for its R-vector spacestructure; recall that there is a direct sum decomposition VC = V 1,0 ⊕ V 0,1.Let WR = HomR(V,R) be its dual and WC = HomR(W,C); WC inherits adirect sum decomposition WC = W 1,0 ⊕W 0,1.

Lemma 2.14. There is a 1-to-1 correspondence between Hermitian formsV × V → C and elements of W 1,1

R given by h 7→ ω = −=h. If ω ∈ W 1,1R ,

h(·, ·) = ω(·, I·) is Hermitian, where I is the complex structure endomor-phism.

Proof. Let h = <h+i=h be a Hermitian form and ω = −=h; for all u, v ∈ V ,

h(u, v) = <h(u, v) + i=h(u, v) = h(u, v) = <h(v, u) + i=h(v, u),

so that ω(u, v) = −ω(v, u) is an alternating real form. We now show thatω ∈ W 1,1, i.e. that ω(u, v) = 0 for all u, v ∈ V 1,0 or u, v ∈ V 0,1. Note thatsince ω is a real form and for all v ∈ V 1,0, v ∈ V 0,1, it is enough to checkthat ω(u, v) = 0 for all u, v ∈ V 1,0. We have seen that elements of V 1,0 areof the form v − iI(v) for v ∈ V ; consider for u, v ∈ V :

ω(u− iI(u), v− iI(v)) = ω(u, v)−ω(I(u), I(v))− i(ω(u, I(v)) +ω(I(u), v)).

Since h(I(u), I(v)) = ih(u, I(v)) = −i2h(u, v) = h(u, v) and h(u, I(v)) =−h(I(u), v), we have ω(u, v) = ω(I(u), I(v)), and ω(u, I(v)) = −ω(I(u), v).This finishes the proof.

38

The isomorphism of Lemma 2.14 does not depend on the choice of abasis for V , but it is helpful to see its expression in a specific basis ofV . Recall that if v1, · · · , vn is a basis for V , we write v∗1, · · · , v∗n for theassociated dual basis. If h = Σhj,kv

∗j ⊗ v∗k, then the associated (1, 1)-form is

ω = i2Σhj,kv

∗j ∧ v∗k.

The form ω is said to be positive if the associated hermitian form h ispositive-definite.

Definition 2.15. Let X be a complex manifold. A Hermitian structureon X is a Hermitian metric h on the complex vector bundle TX,C. TheHermitian manifold (X,h) is Kahler if the real (1, 1)-form ω associated toh is closed for the exterior differential, i.e. if dω = 0.

Example 2.16. 1. The real (1, 1)-form ω = i2Σdzi ∧ dzi associated to

the standard Hermitian structure on Cn is Kahler.

2. Let X be a complex curve; then any Hermitian structure on X isKahler.

3. Let Λ ⊂ Cn be a lattice of rank 2n. The complex torus X = Cn/Λ is aKahler manifold. Let ω′ = i

2Σhj,kdzj ∧ dzk be a (1, 1)-form defined onCn, and assume that ω′ has constant coefficients. Then ω′ is invariantunder the action of Λ, and hence ω′ = π∗ω for ωinΩ1,1

X ∩ Ω2X,R, where

π : Cn → X. The form ω is Kahler.

Let h be a Hermitian form on a C-vector space V ; then g = <h definesan Euclidian inner product on TX,R. In particular, any Hermitian manifoldis also a Riemannian manifold. Assume that (X,h) is a Hermitian manifold;we have seen that X has a canonical orientation. Recall that if v1, · · · , vnis a basis of the holomorphic tangent bundle TX , there is a canonical iso-morphism TX = (TX ,C)1,0 and v1, I(v1), · · · , vn, I(vn) is an oriented basisof TX ,R over R (Exercise 1, Example Sheet 2).

Since (X,h) is in particular a Riemannian manifold, it has a canonicalvolume form, that is a nowhere vanishing section of Ω2n

X,R whose value atP ∈ X is the unique form that is strictly positive on all oriented bases ofTX,P,R and has norm 1 for the induced metric on Ω2n

X,P,R.

Lemma 2.17. The volume form associated to the Hermitian metric h onX is ωn

n! .

Proof. Let z1, · · · zn be local holomorphic coordinates near P such that∂∂z1

, · · · , ∂∂zn

is an isometric frame of TX,C near P , i.e. hP ( ∂∂zi, ∂∂zj

) = δi,j ,

39

then if <(zi) = xi and =(zi) = yi,∂∂x1

, ∂∂y1

, · · · , ∂∂xn

, ∂∂yn

is an oriented basisof TX,P,R. The volume form of (X,h) is then the unique form with value 1on ∂

∂x1∧ ∂∂y1∧ · · ·∧ ∂

∂xn∧ ∂∂yn

. It is enough to prove that ωn

n! ( ∂∂x1∧ ∂∂y1∧ · · ·∧

∂∂xn∧ ∂∂yn

) = 1.

By definition of the coordinates z1, · · · , zn near P ∈ X, ωP = Σnj=1dzj ∧

dzi; so that ωnP = ( i2)nΠnj=1dzj ∧ dzj and the result follows from dzj ∧ dzj =

(dxj + idyj) ∧ (dxj − idyj).

When X is compact, this implies that∫X

ωn

n! > 0. We get as a corollary:

Lemma 2.18. Let (X,ω) be a compact Kahler manifold, the form ωk is notexact for k = 1, · · · , n.

Proof. Assume that ωk = dη, then ωn = ωn ∧ dη = d(ωn−k ∧ η). By Stokestheorem, ∫

Xωn =

∫Xd(ωn−k ∧ η) = 0 :

this is a contradiction.

Remark 2.19. For each k = 1, · · · , n, [ωk] is a nonzero class in H2k(X,R),

and ωk is a section of Ak,kX .

We now come to a useful characterisation of Kahler metrics. Let P ∈ Xand z1, · · · , zn holomorphic coordinates centred at P . Assume that in thesecoordinates, ω = i

2Σhjk(z)dzj ∧ dzk. The Kahler condition ∂ω = 0 writes:

∂hjk∂zl

=∂hlk∂zj

for all 1 ≤ j, k, l ≤ n (5)

Theorem 2.20. Let (X,ω) be a Hermitian manifold. Then (X,ω) is Kahlerif and only if for all P ∈ X there are local holomorphic coordinates z1, · · · , zncentred at P ∈ X such that in these coordinates, hjk(z) = δjk+O(|z|2). Sucha system of coordinates is called normal for the Kahler form ω.

Proof. If the metric has the given expression, it is clear that it is Kahler.We prove that if ω is Kahler, there are such local holomorphic coordinatesnear each P ∈ X. As is noted in Remark 2.12, we may pick holomorphiccoordinates z1, · · · , zn such that hjk(z) = δjk +O(|z|). Consider the Taylorexpansion of hjk to the first order:

hjk(z) = δjk + Σ(ajklzl + a′jklzl) +O(|z|2),

40

where ajkl =∂hjk∂zl

and a′jkl =∂hjk∂zl

. We want to find a system of holomorphic

coordinates ζ1, · · · , ζn such that∂hjk∂ζl

=∂hjk∂ζl

= 0. Since hjk = hkj , akjl =

a′jkl, and (5) implies that ajkl = alkj . Define local holomorphic coordinates:

ζk = zk +1

2Σajklzjzj ,

thendζk = dzk + Σajklzldzj ,

and ω = iΣdζk ∧ dζk +O(|ζ|2).

This shows that in the neighbourhood of each point, a Kahler metric isisomorphic to a constant metric to the first order, ω osculates the standardHermitian metric on Cn to order 2.

Remark 2.21. The holomorphic tangent bundle TX , viewed as a complexvector bundle over the differentiable manifold X, is canonically isomorphicto TX,R. When X is Kahler, the matrix of the Levi-Civita connection of(TX,R, g) coincides with the matrix of the Chern connection of (TX , h), whereg = <h. For more on this point of view, see [Voi02].

2.3 Examples of Kahler manifolds

Chern classes of Line bundles Let X be a compact complex manifoldand L→ X a complex line bundle over X. Recall that if D is a connectionon E, the curvature ΘD ∈ A2(X,EndE) is a global section of the sheafA2(EndE). Here, since E = L is a line bundle,

EndL = L∗ ⊗ L ' X × C,

and ΘD is in fact a global 2-form. Fix a local frame U, e1 for L → Xover U ⊂ X and write in this frame D = d+A, where A is a 1-form. ThenΘD = dA is a closed 2-form. Consider [ΘD] ∈ H2(X,C) the correspondingDe Rham cohomology class.

Claim 2.22. The class [Θ] = [ΘD] ∈ H2(X,C) is independent of the choiceof connection D on L→ X.

Proof. Let D′ be another connection on L → X that is represented in thelocal frame U, e1 by the 1-form A′. For any s ∈ A0(L), (D − D′) · s =(A−A′) ·s. Using the transition relations for A,A′, we see that A−A′ gluesto a global 1-form B ∈ A1(X). Then, ΘD −Θ′D = dB, and [ΘD] = [ΘD′ ] ∈H2(X,C).

41

Proposition 2.23. Let L→ X be a Hermitian line bundle over a differen-tiable manifold. The first Chern class of L is

c1(L) =[ i2π

ΘD

]∈ H2(X,R),

where D is an arbitrary connection on L.

Proof. We have seen that c1(L) is a well defined De Rham class in H2(X,C),we show that it is real. Being real or complex valued is a local property, so wecan work in a local trivialisation U, e1 of L. In that frame, the connectionD = d + A, for A ∈ A1(EndL), and ΘD = dA. Since we may assume thatthe frame is isometric, and the connection is Hermitian −At = −A = A andiΘD = −idA = iΘD.

Remark 2.24. The constant i2π actually ensures that c1(L) ∈ H2(X,Z).

Remark 2.25. The Lefschetz Theorem on (1, 1)-classes (Example Sheet 4)states that if ω is a d-closed section of A1,1(X) with [ω] ∈ H2(X,Z), thenω = c1(L) for some holomorphic Hermitian line bundle L→ X.

Remark 2.26. If E → X is a complex vector bundle, define c1(V ) =c1(detV ). If L is any line bundle, c1(E ⊗ L) = c1(E) + rkE · c1(L).

Assume that L → X is a holomorphic line bundle, and let DL be theChern connection on L. The metric h defines a differentiable functionh : X → R∗+ with h(z) = |e(z)|2, in a holomorphic local frame U, e. Thereis a well defined weight function ϕ : U → R such that ϕ(z) = 1/ log h(z).The Chern connection DL is given in the local holomorphisc frame U, eby A = H

−1∂H, i.e. by A = −∂ϕ, and

ΘL = ∂∂ϕ = −∂∂ϕ.

It is clear that i2πΘL is represented by a real valued, d-closed (1, 1)-form ωL.

By Lemma 2.14, there is a Hermitian form hL associated to ωL.

Definition 2.27. The holomorphic line bundle L → X is positive if hLdefines a Hermitian metric on TX , i.e. if in a holomorphic frame for L→ X,ΘL is given by a weight function ϕ

ΘL = Σ∂2ϕ

∂zj∂zkdzj ∧ dzk,

with ∂2ϕ∂zj∂zk

positive-definite for every z ∈ U .

42

Exercise 2.28. Show that if L → X is globally generated, i.e. if for everyP ∈ X, there is a section σ ∈ Γ(X,L) with σ(P ) 6= 0, then the Hermitian

form hL is defined by a weight function ϕL such that ∂2ϕ∂zj∂zk

is positive

semi-definite.

Exercise 2.29. Show that L is positive if and only if L⊗n is positive for somen ∈ N∗.

A positive holomorphic line bundle L → X defines a Kahler structureon X.

The Fubini-Study metric on Pn Recall that the tautological line bun-dle OPn(−1) → Pn is a subset of Pn × Cn+1. The standard Hermitianproduct on Cn+1 < v,w >= vt · w induces a Hermitian product h on thevector bundle Pn × Cn+1, given by h((x, v), (x,w)) =< v,w >= vt · w.The line bundle OPn(−1) inherits a Hermitian metric, which dualised givesa Hermitian metric on OPn(1). Explicitly, if [z] = [z0: · · · :zn] ∈ Pn ands1, s2 ∈ H0(Pn,OPn(1)) are sections,

hz(s1(z), s2(z)) =< s1(z), s2(z) >

< z, z >.

Recall that on the open set Ui = zi 6= 0 ⊂ Pn, a local trivialisationof OPn(1) is given by the section si ∈ H0(Ui,OPn(1)) with si([z]) = zi.Local holomorphic coordinates are given on Ui by z0, · · · , zi, · · · , zn where[z] = [z0: · · · :zi−1:1:zi+1: · · · :zn]. Then

hz(si([z]), si([z])) =1

1 + Σz2j

,

and the associated hermitian form on Ui is:

ωOPn (1) =1

2iπ∂∂ log hz(si([z]), si([z])) = − 1

2iπ∂∂ log(1 + Σz2

j ).

Lemma 2.30. The form ωOPn (1) is positive. This form is the Fubini Studymetric ωFS on Pn.

Proof. We check that ωFS is positive definite at 0 ∈ Ui (that is at the point[0: · · · :1: · · · :0] ∈ Pn);

∂ log(1

1 + Σz2i

) =∂(1 + Σ|zi|2)

1 + Σ|zi|2=

Σzidzi1 + Σ|zi|2

,

43

hence:

∂∂ log(1

1 + Σz2i

) =(1 + Σ|zi|2)dzi ∧ dzi − Σzidzi ∧ Σzidzi

(1 + Σ|zi|2)2.

At 0 ∈ Ui, ω = i2πΣdzi ∧ dzi is positive. Since ω is invariant under the

transitive action of SU(n+ 1) on Pn, it is positive everywhere on Pn.

Remark 2.31. Note that if π : Cn+1 \ 0 → Pn is the projection map,π∗ωFS = i

2π∂∂ log |z|2 on Cn+1 \ 0.

Assume that Y ⊂ X is a complex submanifold of X, if X is a Kahlermanifold, j∗ωX = ωY is a Kahler form on Y . We then obtain:

Corollary 2.32. Any projective manifold X ⊂ PN is Kahler.

The unexpected partial converse of this statement is the Kodaira em-bedding theorem, which we admit.

Theorem 2.33 (Kodaira embedding Theorem). Let X be a compact Kahlermanifold and L → X a holomorphic line bundle. Then L is positive ifand only if there is a holomorphic embedding φ : X → PN of X into someprojective space PN such that φ∗OPN (1) = L⊗m for some m ∈ N.

Remark 2.34. Note that if there exists such an embedding, the pullbackφ∗OPN (1) = L⊗m defines a form ωL⊗m = φ∗ωFS , which is positive. Con-versely, if X is known to be projective, then by the so called GAGA principle,L is an algebraic line bundle and if L is positive,

∫V c1(L)dimV > 0 for any

irreducible subvariety V ⊂ X, and hence, by Nakai’s criterion, L is am-ple. The truly astonishing statement contained in the Kodaira EmbeddingTheorem is that the positivity of a line bundle on a compact Kahler man-ifold guarantees the existence of enough holomorphic sections to define anembedding.

Projective bundles Let π : E → X be a rank r + 1 holomorphic vectorbundle over a complex manifold X. Define the projective bundle P(E) =(E \ 0-section)/C∗ → X. If E|Ui ' Ui × Cr+1 is a local trivialisa-tion of E, P(E)|Ui ' Ui × Pr, and over Ui ∩ Uj , the identification of thetrivialisations P(E)|Ui ' P(E)|Uj is given by hi,j : z → P GL(r,C), wherehi,j : z → GL(r,C) is the cocycle of E.

Lemma 2.35. If X is a compact Kahler manifold, P(E) is Kahler.

44

Proof. Let S ⊂ π∗E → E be the tautological line bundle, i.e. the holomor-phic line bundle with fibre over (x, [V ]) ∈ P(E) the 1-dimensional subspaceV ⊂ Ex. Similarly to the case of Pn, the line bundle OP(E)(1) is then definedas the dual S∗. On a fibre of π, by definition,

OP(E)(1)|π−1(x) ' OP(Ex)(1) ' OPr(1).

A Hermitian metric h on E induces a Hermitian metric on π∗E and henceon S and S∗. The associated Chern form ωE is positive on each fibreπ−1(x) = P(Ex), because, there ωEπ−1(x) ' ωFSP(Ex). However, this formis not necessarily positive globally. Let ωX be a Kahler form on X, then onU ⊂ X there is a constant λU >> 0 such that ωλU = ωE + λUωX is positiveon P(E)|U . If X is compact, there is a constant λ >> 0 such that ωλ ispositive everywhere, and P(E) is Kahler.

Remark 2.36. Note that P(E) can be Kahler without X being Kahleritself.

Blowups Let Y ⊂ X be a complex submanifold of codimension k. ByTheorem 1.9, near each point P ∈ Y , there are local holomorphic coordinatesz1, · · · , zn on X such that Y = z1 = · · · = zk = 0 ⊂ X. We will define theblowup of X along Y :

σ : XY → X.

The blowup σ is a proper holomorphic map σ from a complex manifold XY

such that:

• σXY

: XY r σ−1(Y ) ' X r Y ,

• E = σ−1(Y ) ⊂ XY is a smooth hypersurface and σ|E : E → Y isisomorphic to the natural projection P(NY/X)→ Y .

Remark 2.37. Blowups are very important in Algebraic Geometry; theyare the simplest example of birational maps. In fact, if X is an algebraicvariety and Y ⊂ X is a subvariety, the blowup of X along Y has the followinguniversality property: if f : Z → X is a morphism such that f−1(Y ) is aCartier divisor, then f factors through σ.

Let U ⊂ X be an open set and let z1, · · · , zn be holomorphic coordinateson U such that Y ∩ U = z1 = · · · = zk = 0 ⊂ U . Define σ|U : UY ⊂Pk−1 × U → U , where UY = ([Z], z) ∈ Pk−1 × U |Zizj = ziZj and σ is the

45

projection on the second factor. The map σ|U is clearly proper, and is an

isomorphism over U r (U ∩ Y ). For y ∈ Y , σ−1(y) ' Pk−1.Recall that if Ui, φi is a complex atlas of X, the holomorphic tan-

gent bundle TX is defined by the cocycle Ui, J(φi,j) φj, where J isthe holomorphic Jacobian and φi,j = φi φ−1

j are the transition func-

tions. If Yi⊂ X is a complex submanifold, that corresponds to the embed-

ding i : (zk, · · · , zn) 7→ (0, · · · , 0, zk, · · · , zn), the holomorphic vector bundlei∗TX = TX |Y is defined by the cocycle Ui ∩ Y, J(φi,j) φ−1

j i. There isan exact sequence of vector bundles

0→ TY → TX |Y → N ∗Y/X → 0

which defines the normal bundle NY/X .

We identify the local constructions UY → Y and VY → V over U ∩V bysetting the biholomorphic map

σ−1U (U ∩ V ) ' σ−1

V (U ∩ V )

to be ([Z], z) 7→ (MUV [Z], z), where MUV ∈ P GL(k−1,C) are the transitionmatrices of the cocycle NY/X .

Note that if Y ∩U = z1 = · · · zk = 0 and V ∩Y = f1 = · · · = fk = 0for holomorphic functions (with linearly independent differentials), then byconstruction of NY/X , over U ∩ V ∩ Y :

(f1, · · · , fk) = MUV (z1, · · · , zk) and (df1, · · · , dfk) = MUV (dz1, · · · , dzk).

The biholomorphic identification over U ∩V shows that XY → X is welldefined (the gluings satisfy the cocycle conditions..) and that over Y , σcoincides with P(NY/X)→ Y .

Remark 2.38. Note that when Y = σ = 0 ⊂ X is a smooth hypersurface(dσ is injective), the blowup XY → X is an isomorphism.

Example 2.39. Recall the construction of the tautological line bundleOPn(−1) ⊂ Pn×Cn+1. The projection to the second factorOPn(−1)→ Cn+1

is the blowup of Cn+1 at the origin 0 ∈ Cn+1. Note that N0/Cn+1 ' Cn+1.

Claim 2.40. E = σ−1(Y ) is a smooth hypersurface in XY .

Denote [Z] = [Z0: · · · :Zk] and y = (y1, · · · , yn) = (0, · · · , 0, zk+1, · · · , zn),for ([Z], y) ∈ E; there is an index i0 such that Zi0 6= 0. A local holomorphicequation for E near ([Z], y) is given by fi0 : ([Z], y) 7→ yi0 .

46

Claim 2.41. There is a holomorphic line bundle L → XY that is trivialoutside E and such that L|E ' OP(NY/X)(1).

Ex. 5 in Example Sheet 3 constructs a line bundle OXY

(E) associated

to a smooth hypersurface E ⊂ XY , Ex. 4(ii) in Example Sheet 2 showsthat N

E/XY' O

XY(E) ⊗ OE = O

XY(E)

|E. It is enough to prove that

NE/XY

' OP(NY/X)(−1). This is clear on the explicit construction of XY –

this is the same idea as in Remark 2.38.

Proposition 2.42. Let X be a Kahler manifold and Y ⊂ X be a compactsubmanifold. The blowup XY is Kahler and compact if X is compact.

Proof. The construction of σ shows that σ is proper, and hence XY is com-pact if X is. Let ωX be a Kahler form on X; the pullback σ∗ωX is a closed(1, 1)-form on XY and it is positive at P ∈ XY that does not belong to E.The form σ∗ωX is only semi-definite on E; indeed on E, the tangent vectorslying in the tangent spaces of fibres of σ are isotropic for σ∗ωX . Let ω′ = ωLbe the (1, 1)-form associated to the holomorphic line bundle in Claim 2.41.Then dω′ = 0, ω′ = 0 outside a compact neighbourhood of E and ω′ positivedefinite on E and hence on the fibres of σ. Since Y is compact, there thenexists a constant C >> 0 such that Cσ∗ωX + ω′ is positive.

3 Hodge Theory on Hermitian manifolds

In this section, we relate De Rham cohomology classes to harmonic forms ona Hermitian manifold (X,h). On the one hand, we have seen that De Rhamcohomology coincides with Betti cohomology; this shows that De Rham co-homology classes really are topological objects. On the other hand, harmonicforms are solutions of PDEs that depend on the Riemannian/Hermitianstructure. Hodge Theory shows that each cohomology class is representeduniquely by a harmonic form. This describes topological data on a manifoldin terms of very explicit geometric objects.

3.1 Differential Operators

Let X be a compact oriented Riemannian manifold and (E, gE) an Euclidianvector bundle. Recall that we may define a L2-metric on A0

c(E) the spaceof compactly supported sections of E by:

(s1, s2)L2 =

∫XgE(s1, s2) volX .

47

If E,F are two Euclidian vector bundles on X and if P : A0c(E)→ A0

c(F ) is alinear operator, then we may define the formal adjoint P ∗ : A0

c(F )→ A0c(E)

by the requirement, for all s1 ∈ A0c(E) and s2 ∈ A0

c(F ):

(Ps1, s2)L2 = (s1, P∗s2)L2 .

Let X be a compact Riemannian manifold with dimRX = m, the Hodgeoperator ? : AkX,R → A

m−kX,R is defined by α∧?β =< α, β > volX , where <,>

is the metric induced on Ω∗X by the Riemannian metric.

If ∂∂x1

, · · · , ∂∂xm

is an isometric (local) frame for the metric on TX,R,dxi1 ∧ · · · ∧dxik ; i1 < · · · < ik is an isometric frame for the induced metricon Ωk

X , and the Hodge operator is then defined by linearity and permutationsby imposing

?(dx1 ∧ · · · ∧ dxk) = dxk+1 ∧ · · · ∧ dxn.

In particular ?1 = vol, ? vol = 1 and ?? = (−1)k(m−k) on AkX .

The adjoint d∗ : Ak+1X,R → AkX,R of d : AkX,R → A

k+1X,R satisfies:

d∗ = (−1)mk+1 ? d ? . (6)

Assume now that (X,h) is a compact Hermitian manifold; recall thatthe underlying differentiable manifold X is orientable and endowed witha Riemannian structure <h = g. We extend the previous notions by C-linearity. Let n = dimCX, the Hodge operator ? : AkX → A

2n−kX is such that

for all α, β ∈ AkX , α ∧ ?β =< α, β > vol. In particular, it is clear from thedefinition that ? : Ap,qX → An−q,n−pX is a sheaf isomorphism, and as above,we check that ?1 = vol, ? vol = 1 and ?? = (−1)k(2n−k) on AkX .

For any Hermitian vector bundle E, we define an associated L2-metricon A0

c(E) as above.

When X is compact, the space of differential forms (resp. the space ofk-forms) admits a direct sum decomposition A∗X =

⊕2nk=1AkX (resp. AkX =⊕k

p=1Ap,k−pX ) that is orthogonal for the L2-metric induced by the Hermitian

structure:

(α, β)L2 =

∫Xh(α, β) vol =

∫Xα ∧ ?β.

Each Ap,qX is an infinite-dimensional vector space endowed with a scalarproduct; Lp,q denotes its completion with respect to the associated L2-norm(·, ·)L2 .

Exercise 3.1. Check that if hk and hp,q are the restrictions of the Hermitianmetric to Ωk

X,C and Ωp,qX respectively, 2khk = Σhp,q.

48

Since X is a complex manifold, the exterior differential splits as a sumd = ∂ + ∂ and from (5),

d∗ = − ? d ? .

Definition-Lemma 3.2. Let X be a compact Hermitian manifold. Theoperators

∂∗ = − ? ∂? : Ap+1,qX → Ap,qX and ∂

∗= − ? ∂? : Ap,q+1

X → Ap,qXare formal adjoints for the operators ∂ and ∂ respectively. They satisfyd∗ = ∂∗ + ∂

∗, (∂∗)2 = (∂

∗)2 = 0 and ∂

∗∂∗ = −∂∗∂∗.

Proof. It is enough to prove that for any α ∈ Ap,qX and β ∈ Ap,q+1X , (∂α, β)L2 =

(α, ∂∗β)L2 . We have

(∂α, β)L2 =

∫X∂α ∧ ?β,

and since α ∧ ?β is of type (n, n− 1),

d(α ∧ ?β) = ∂(α ∧ ?β) = ∂α ∧ ?β + (−1)p+qα ∧ ∂(?β).

Since ? is a real operator, ∂(?β) = ∂ ? β = (−1)p+q ? ?∂ ? β and the resultfollows from the Stokes’ formula.

Definition 3.3. The Laplace operators associated to d, ∂ and ∂ are:

∆d = dd∗ + d∗d , ∆∂ = ∂∂∗ + ∂∗∂ , and ∆∂ = ∂∂∗

+ ∂∗∂.

Remark 3.4. Note that while ∆∂ and ∆∂ are bihomogeneous operators,i.e. they are operators Ap,qX → A

p,qX that preserve both the degree of forms

and their types, ∆d : AkX → Ak+1X does not necessarily preserve the type of

k-forms.

Since we assume that X is compact, if α ∈ AkX ,

(α,∆dα)L2 = (dα, dα)L2 + (d∗α, d∗α)L2 = ||dα||L2 + ||d∗α||L2

(α,∆∂α)L2 = ||∂α||L2 + ||∂∗α||L2

(α,∆∂α)L2 = ||∂α||L2 + ||∂∗α||L2

Definition 3.5. A d-harmonic (resp. ∂-harmonic, ∂-harmonic) is a formα ∈ ker ∆d (resp. ker ∆∂ , ker ∆∂). When X is compact, α is d-(resp. ∂,∂-)harmonic when α ∈ ker d ∩ ker d∗ (resp. α ∈ ker ∂ ∩ ker ∂∗, α ∈ ker ∂ ∩ker ∂

∗). We denote Hkd(X,h),Hk∂(X,h) and Hk

∂(X,h) the spaces of d, ∂ and

∂-harmonic forms on X.

49

From what we have seen, the spaces Hk∂(X,h) and Hk∂(X,h) admit direct

sum decompositions:

Hk∂(X,h) =⊕p+q=k

Hp,q∂ (X,h) and Hk∂(X,h) =

⊕p+q=k

Hp,q∂

(X,h),

but since the Laplacian ∆d is not bihomogeneous, no such decompositionexists a priori on Hkd(X,h).

Properties of harmonic spaces on Hermitian manifolds Recall thatthe Hodge operator ? induces a C-linear isomorphism ? : AkX → A

2n−kX ; from

the definition of d∗, this is an isomorphism ? : Hkd(X,h)→ H2n−kd (X,h), and

more specifically ? : Hp,qd (X,h)→ Hn−p,n−qd (X,h). From the definition of ∂∗

and ∂∗, we see that

? : Hp,q∂

(X,h)→ Hn−p,n−q∂ (X,h)

is an isomorphism. Complex conjugation interchangesHp,q∂

(X,h) andHq,p∂ (X,h).Assume that (X,h) is a compact, connected Hermitian manifold. The

pairing Hp,q∂

(X,h)×Hn−p,n−q∂

→ C defined by

(α, β)→∫Xα ∧ β

is non degenerate because (α, ?α) > 0 for α 6= 0; this establishes SerreDuality for harmonic forms, that is:

Hp,q∂

(X,h) ' Hn−p,n−q∂

(X,h)∗, (7)

where Hn−p,n−q∂

(X,h)∗ denotes the dual vector space.

Elliptic Operators

Definition 3.6. Let E,F be two complex vector bundles and P : A0(E)→A0(F ) be a C-linear morphism of sheaves. P is a differential operator oforder k if for any simultaneous trivialisation E|U ' U×Cp, and F|U ' U×Cqover U ⊂ X,

P (s1, · · · , sp) = (t1, · · · , tq), where tj = ΣPI,i,j∂si∂xI

,

and PI,i,j = 0 for any multi-index |I| > k and PI,i,j 6= 0 for some I with|I| = k.

50

The operator P decomposes as a sum P = P1 + · · ·+ Pk, where each Piis a section of Hom(E,F )⊗ Symi TX .

The symbol of P is σP = Pk. For x ∈ X, σP (x) ∈ Hom(Ex, Fx) ⊗Symk TX , i.e. is a degree k homogeneous map ΩX,x → Hom(Ex, Fx). Theoperator P is elliptic if for all x ∈ X, x 6= 0, σP (x) is injective.

We admit the following.

Lemma 3.7. The Laplacian operators ∆d,∆∂ and ∆∂ are elliptic and self-adjoint. Their symbols are

σd(α) = −||α||2L2 · Id , and σ∂(α) = σ∂(α) = −1

2||α||2L2 · Id .

Proof. This is an easy computation in a local holomorphic coordinate systemfor X.

We admit the following Fundamental Theorem on elliptic operators.

Theorem 3.8. Let X be a compact complex manifold and E,F → X betwo Hermitian vector bundles with rkE = rkF . Let P : E → F be an el-liptic differential operator, then kerP ⊂ A0(E) has finite dimension andP (A0(E)) ⊂ A0(F ) is closed and has finite codimension.There is a decom-position as a direct sum:

A0(E) = kerP ⊕ P ∗(A0(F ))

that is orthogonal for the L2 metric (induced by the metric on E).

Corollary 3.9. Let (X,h) be a compact Hermitian manifold. Then there isa orthogonal decomposition:

Ak(X) = Hkd(X,h)⊕ dAk−1(X)⊕ d∗Ak+1(X),

and Hkd(X,h) has finite dimension. Similarly,

Ap,q(X) = Hp,q∂ (X,h)⊕ ∂Ap−1,q(X)⊕ ∂∗Ap+1,q(X),

Ap,q(X) = Hp,q∂

(X,h)⊕ ∂Ap,q−1(X)⊕ ∂∗Ap,q+1(X)

and Hp,q∂ (X,h) and Hp,q∂

(X,h) have finite dimension.

Remark 3.10. The crucial point is the existence of such a decomposition(orthogonality is then easy). For a proof of this theorem and more onHarmonic Theory, see [Dem96].

51

Corollary 3.11. Let (X,h) be a compact Hermitian manifold. Then, theprojections

Hkd(X,h)→ HkDR(X,C) and Hp,q

∂(X,h)→ Hp,q(X)

are isomorphisms.

Proof. We prove the statement for the operator ∂ (the proof for d is identi-cal). Let α ∈ Hp,q

∂(X,h) be a ∂- harmonic form; since α is ∂-closed, α defines

a Dolbeault cohomology class [α] ∈ Hp,q(X). Assume that β ∈ Ap,q(X) is a∂-closed form. By Corollary 3.9,

β = α+ ∆∂γ,

where α ∈ Hp,q∂

(X,h) and γ ∈ Ap,q(X). since ∂β = ∂α = 0, ∂(∂∗∂γ) = 0,

so that ∂∗(∂γ) ∈ ker ∂ ∩ im ∂

∗. By Corollary 3.9, ker ∂ ∩ im ∂

∗= 0, so

that ∆∂γ = ∂∂∗γ is ∂-exact. This shows that [β] = [α], and the projection

Hp,q∂

(X,h)→ Hp,q(X) is an isomorphism.

Remark 3.12. Corollary 3.11 shows that every De Rham cohomology class(resp. Dolbeault cohomology class) has a unique d-harmonic (resp. ∂-harmonic)representative.

3.2 The case of Kahler manifolds

We will show that if (X,h) is a Hermitian manifold and if the metric isKahler–we have seen that this is a local condition–then the various Laplaceoperators are proportional, so that d-harmonic forms are ∂ and ∂-harmonic.By definition, a Kahler metric coincides with the standard Hermitian metricon Cn up to order 2; this will imply relations between linear and differentialoperators on X.

In this section, we assume that (X,h) is a Kahler manifold, and wedenote ω its Kahler form.

Definition-Lemma 3.13. The Lefschetz operator is the linear operatorAkX → A

k+2X defined by

α 7→ ω ∧ α.Its formal adjoint Λ: Ak−2

X → AkX satisfies Λ = (−1)k ? L?.

Proof. The only thing there is to prove is that (Lα, β)L2 = (α, (−1)k?L?β)L2

for α ∈ Ak(X) and β ∈ Ak+2(X). For any x ∈ X,

Lα ∧ ?β = ω ∧ α ∧ ?β = (−1)kα ∧ ? ? ω ∧ ?β = (−1)kα ∧ ?(?ω ∧ ?β)

because ω is a real form.

52

Recall that for any differential operators P,Q of degrees p, q, the Liebracket

[P,Q] = P Q−Q P

is a differential operator of degree p+ q.

Proposition 3.14 (The Kahler identities). Let (X,ω) be a Kahler manifold,then:

1. [Λ, ∂] = −i∂∗,

2. [Λ, ∂] = i∂∗,

3. [∂∗, L] = i∂,

4. [∂∗, L] = −i∂.

Proof. Note that the first two equalities imply the third and fourth by ad-junction. Also the first implies the second by complex conjugation becauseL and Λ are real operators and hence for any α ∈ AkX ,

[Λ, ∂](α) = [Λ, ∂](α).

We therefore just need to prove the first equality. Note that since Λ is anoperator of degree 0, [Λ, ∂] has degree 1 and hence it is enough to prove iton Cn equipped with the standard Hermitian metric by Theorem 2.20. Thisis the content of Lemma 3.15.

Lemma 3.15. Let U ⊂ Cn be an open set endowed with the standard Her-mitian metric, i.e. ω = iΣdzj ∧ dzj. Then [∂

∗, L] = i∂.

Recall that if θ ∈ A0(TX,C) is a vector field on X, the interior productwith θ is the map

i(θ) : AkX → Ak−1X , (8)

such that for any vector fields η1, · · · , ηk−1 ∈ A0(TX,C) and for any k-formα ∈ AkX ,

i(θ)(α)(η1, · · · , ηk−1) = α(θ, η1, · · · , ηk−1).

It follows immediately that if u, v ∈ A•X are two differential forms:

i(θ)(u ∧ v) = i(θ)(u) ∧ v + (−1)deg uu ∧ i(θ)(v).

If θ = θ1,0+θ0,1 is the decomposition of the vector field θ into its componentsof type (1, 0) and (0, 1), the interior products with θ1,0 and θ0,1 define maps:

i(θ1,0) : Ap,qX → Ap−1,qX and i(θ0,1) : Ap,qX → A

p,q−1X .

53

Recall that if z1, · · · , zn are local holomorphic coordinates onX, ∂∂z1

, · · · , ∂∂zn

and ∂∂z1

, · · · , ∂∂zn are local frames for T 1,0

X and T 0,1X respectively. If I, J

are multi-indices with |I| = p and |J | = q, we have:

i(∂

∂zi)(dzI ∧ dzJ) = 0 if i 6∈ I

= (−1)l−1dzI−i ∧ dzJ for I = i1 < · · · < il = i < · · · < ip

and:

i(∂

∂zj)(dzI ∧ dzJ) = 0 if j 6∈ J

= (−1)p+l−1dzI ∧ dzJ−j for J = j1 < · · · < jl = j < · · · < jq

For a k-form α ∈ AkX (on a differentiable manifold X),

d∗α = −Σnl=1Σ|J |=k

∂αJ∂xl

i(∂

∂xl(dxJ). (9)

For a complex manifold X, (9) implies that if α = ΣI,JαI,JdzI ∧ dzJ , then:

∂∗α = −Σnl=1ΣI,J

∂αI,J∂zl

i(∂

∂zl)(dzI ∧ dzJ)

and

∂∗α = −Σn

l=1ΣI,J∂αI,J∂zl

i(∂

∂zl)(dzI ∧ dzJ).

Proof of Lemma 3.15. Fix α ∈ Ap,qX , then:

[∂∗, L](α) = −Σn

l=1i(∂

∂zl)∂(

∂ω∧ α)zl + ω ∧ Σn

l=1i(∂

∂zl)(∂α

∂zl).

Since X = Cn with the standard hermitian structure, ∂(ω∧α)∂zl

= ω ∧ ∂α∂zl

, and

i(∂

∂zl)(ω ∧ ∂α

∂zl) = i(

∂

∂zl))(ω) ∧ ∂α

∂zl+ ω ∧ i( ∂

∂zl)(∂α

∂zl).

Since i( ∂∂zl

)(iΣdzj ∧ dzj) = −idzl,

[∂∗, L](α) = iΣn

l=1dzl ∧∂α

∂zl= i∂α.

54

Theorem 3.16. Let (X,ω) be a Kahler manifold. The Laplace operators∆d,∆∂ and ∆∂ associated to d, ∂ and ∂ satisfy:

∆d = 2∆∂ = 2∆∂ .

Proof. We use the Kahler identities of Proposition 3.14 to express the d-Laplacian in terms of the ∂-Laplacian. Write:

∆d = dd∗ + d∗d = (∂ + ∂)(∂∗ + ∂∗) + (∂∗ + ∂

∗)(∂ + ∂)

= (∂ + ∂)(∂∗ − i[Λ, ∂]) + (∂∗ − i[Λ, ∂])

= ∆∂ − i∂[Λ, ∂]− i∂[Λ, ∂] + ∂∗∂ − i[Λ, ∂]− i[Λ, ∂]∂

Using that ∂∂∗ = i∂[Λ, ∂] = i∂Λ∂ and ∂∗∂ = −i∂Λ∂, and ∂∂ = −∂∂,∂∗∂

∗= −∂∗∂∗, the expression above reduces to:

∆d = ∆∂ + i∂[Λ, ∂] + i[Λ, ∂]∂ = 2∆∂ .

The proof of ∆d = 2∆∂ is similar.

Corollary 3.17. If X is Kahler, the d-Laplacian is bihomogeneous–i.e. ifω ∈ Ap,qX , ∆d(ω) ∈ Ap,qX . If α is a d-harmonic form and if α = Σαp,q is thedecomposition of α into forms of type (p, q), then for each (p, q), αp,q is ad-harmonic form.

Proof. Since ker ∆d = ker ∆∂ = ker ∆∂ , d-harmonic forms are ∂ and ∂-harmonic, the result then follows (see Remark 3.4).

The properties of ∂-harmonic forms on X, and the identification betweend, ∂ and ∂-harmonic forms then imply the Hodge Decomposition for compactKahler manifolds:

Theorem 3.18 (Hodge Decomposition). Let X be a compact Kahler man-ifold, there is a decomposition:

Hk(X,C) '⊕p+q=k

Hp,q(X)

such that Hp,q(X) ' Hq,p(X).

Remark 3.19. The isomorphism between the Betti cohomology group andthe direct sum of the Dolbeault cohomology groups in the above statementreflects the fact that this follows from the isomorphisms between the coho-mology groups and harmonic forms; these depend on the choice of metric apriori.

55

Proposition 3.20. The Hodge decomposition Hk(X,C) =⊕

p+q=kHp,q(X)

does not depend on the choice of Kahler metrics on X. The isomorphismHp,q(X) ' Hq,p(X) does not depend on the choice of Kahler metrics.

Proof. Let Kp,q ⊂ Hk(X,C) be the set of De Rham classes that can berepresented by a form of type (p, q) on X. Since the Dolbeault classes areidentified with ∂-harmonic forms of type (p, q), Hp,q(X) ⊂ Kp,q. We wantto show the reverse inclusion. Let ω be a closed (p, q)-form correspondingto a class [ω] ∈ Kp,q. By Corollary 3.9,

ω = αp,q + ∆dη,

for αp,q a harmonic form of type (p, q) and η ∈ Ap,qX . Since dω = 0, dd∗dη =0, and d∗dη ∈ ker d∩im d∗. By Corollary 3.9 again, we obtain that d∗dη = 0,so that [ω] = [αp,q] which is canonically a class inHp,q(X). The identificationKp,q = Hp,q(X) does not depend on the choice of Kahler metric.

Similarly, since Kp,q = Kq,p, the second assertion holds.

The existence of a unique harmonic representative in each (Dolbeault orDe Rham) cohomology class also implies:

Corollary 3.21. If a (Dolbeault or De Rham) cohomology class on X isrepresentable by a form of type (p, q) and by a form of type (p′, q′) for (p, q) 6=(p′, q′), it is zero.

The properties of harmonic forms explained in Section 3 imply the fol-lowing duality statements on cohomology groups of a Kahler manifold. Themap

α ∈ Hp,q∂

(X) 7→ ?α ∈ Hn−p,n−q∂

defines a duality Hp,q∂

(X) ' (Hn−p,n−q∂

)∗. This implies that the Dolbeaultcohomology groups of X satisfy Serre Duality :

Hp,q(X) ' Hn−p,n−q(X)∗.

The Hodge Decomposition Theorem then implies Poincare Duality :

Hk(X,C) =⊕

Hp,q(X) '⊕

Hn−p,n−q(X)∗ = H2n−k(X,C)∗.

The Hodge Decomposition Theorem imposes conditions on the cohomol-ogy of compact Kahler manifolds.

Corollary 3.22. Let X be a compact Kahler manifold.

56

• For all 0 ≤ p, q ≤ n, hp,q = hq,p = hn−p,n−q = hn−q,n−p,

• bk = Σp+q=khp,q, and in particular, bk is even when k is odd,

• Hk,k(X) 6= (0) for 0 ≤ k ≤ n.

Proof. The first statement follows from the action of complex conjugationand Serre Duality, the second from the Hodge decomposition theorem. Thelast statement is a consequence of Remark 4.44.

Example 3.23. The Hopf surface (see Example sheet 1) is not Kahlerbecause b1(S) = 1. In fact, for a complex compact surface S, it can beshown that S is Kahler precisely when b1(S) is even.

Exercise 3.24. Show thatHp,q(Pn) ' C when 0 ≤ p = q ≤ n, andHp,q(Pn) =(0) otherwise.

The Kahler condition implies that the spaces of harmonic forms for thethree Laplace operators on X coincide. A reformulation of this is the follow-ing lemma, which allows one to identify the Dolbeault cohomology groupswith the Bott-Chern cohomology groups (see ex.2 on Example Sheet 5.),whose definition is independent of the Kahler metric.

Lemma 3.25 (The ∂∂ lemma). Let X be a compact Kahler manifold andα a d-closed form of type (p, q). The following are equivalent:

1. α is d-exact, i.e. α = dβ for β ∈ Ak−1(X),

2. α is ∂-exact, i.e. α = ∂β for β ∈ Ap−1,q(X),

3. α is ∂-exact, i.e. α = ∂β for β ∈ Ap,q−1(X),

4. α is ∂∂-exact, i.e. α = ∂∂β for β ∈ Ap−1,q−1(X),

Proof. It is enough to prove the equivalence of (4) and (2) for instance,because on a Kahler manifold, ∂∂ = −∂∂ and (4) implies (1–3). Assumethat α ∈ Ak(X) is a closed form; since dα = 0, ∂α = ∂α = 0. Assumethat α = ∂β for some form β ∈ Ap−1,q(X). By Corollary 3.9, β = γ + ∆∂εfor some ∂-harmonic form γ and some ε ∈ Ap,q(X). The term ∆∂ε can be

written ∂ε′+∂∗ε” for forms ε′ ∈ Ap,q−1(X) and ε” ∈ Ap,q+1(X). It is enough

to prove that ∂∂∗ε” = 0. Since ∂α = 0, we have ∂∂β = 0. It follows that

∂∂∂∗ε” = 0. We may write ∂∂∂

∗ε” = −∂∂∗∂ε” because X is a complex

manifold, so that ∂∂ = −∂∂ and ∂∗∂ = −∂∂∗. But then the L2-product

(∂ε”, ∂∂∗∂ε”) = ||∂∗∂ε”||2 = 0

57

and ∂∗∂ε” = 0 and ∂∂

∗ε” = 0, so that α = ∂β is of the form ∂(γ + ∂ε′) and

the result follows.

Remark 3.26. The ∂∂-lemma is a direct consequence of the identificationof the spaces of harmonic forms for the three Laplace operators and of thefact that there is a unique harmonic representative for each cohomology classfor De Rham, Dolbeault or Bott-Chern cohomology.

Exercise 3.27. Let X be a Kahler manifold. Show that any holomorphicp-form α ∈ Γ(X,Ωp

X) = Hp,0(X) is harmonic.

3.3 The Lefschetz Decomposition

The De Rham cohomology groups Hk(X,R) of a compact Kahler manifoldX admit another decomposition, which is of a topological nature. TheLefschetz operator L and its adjoint Λ (see 3.13) define linear operators ondifferential forms.

Exercise 3.28. L and Λ commute with the Laplace operator ∆d.

Lemma 3.29. The commutator [L,Λ] restricts to (k − n) Id on AkX .

Proof. Since L and Λ are operators of degree 0, we may assume that (X,ω)is Cn equipped with the standard Hermitian structure in order to computetheir commutator; we therefore consider:

ω =i

2Σnj=1dzj ∧ dzj .

Recall from Definition-Lemma 3.13 that Λ = ?−1L?. On AkX , we have (checkit!)

?−1 (dzj∧) ? = (−1)k+12i(∂

∂zj), and ?−1 (dzj∧) ? = (−1)k+12i(

∂

∂zj).

where i(u) for a vector field u denotes the interior product with u. Thisshows that for every j,

?−1 (i

2(dzj ∧ dzj)∧) ? = −2i · i( ∂

∂zj∧ ∂

∂zj),

and

[L,Λ] = Σj,k[(dzj ∧ dzj)∧, i(∂

∂zk∧ ∂

∂zk)].

58

It is clear that each operator [(dzj ∧ dzj)∧, i( ∂∂zk∧ ∂∂zk

)] is zero unless j = k,so that

[L,Λ] = Σj [(dzj ∧ dzj)∧, i(∂

∂zj∧ ∂

∂zj)].

Any form α ∈ AkX can be written α = ΣI,J,MαI,J,MdzI ∧ dzJ ∧ ωM ,where I, J,M ⊂ 1, · · · , n are disjoint sets of indices, and ωM = dzi1 ∧dzi1 ∧ · · · ∧ dzim ∧ dzim , for M = i1 < · · · < im. The sets I, J,K satisfy|I|+ |J |+ 2|M | = k.

Write K = 1, · · · , nr (I ∪ J ∪M), then

dzl ∧ dzl ∧ (αI,J,MdzI ∧ dzJ ∧ ωM ) = 0 unless l ∈ K , and

i(∂

∂zl∧ ∂

∂zl)(αI,J,MdzI ∧ dzJ ∧ ωM )) = 0 unless l ∈M

. It follows that:

[L,Λ](αI,J,MdzI ∧ dzJ ∧ ωM ) = (|M | − |K|) · αI,J,MdzI ∧ dzJ ∧ ωM ,

and since |M | − |K| = k − n, the result follows.

Lemma 3.30. The operator Ln−k : ΩkX,R → Ω2n−k

X,R is an isomorphism (and

hence Ln−k : AkX → An−kX is an isomorphism of sheaves).

Proof. Since rk ΩkX,R = rk Ω2n−k

X,R , it is enough to prove that Ln−k is injective.

By induction, one shows that [Lr,Λ] = r(k−n)+r(r−1)Lr−1 for all r ≥ 1.If Lrα = 0 for some α ∈ Ωk

X,R, assume that Lr−1 is injective, then since

Lr−1(LΛ− (r(k − n) + r(r − 1)) Id)(α) = 0,

α is of the form Lβ for some β ∈ Ωk−2X with Lr+1β = 0. Conclude by

induction of the degree of α.

Denote Πk : A∗X → AkX be the projection of differential forms on theirdegree k component, and define the operator:

h = Σ2nk=0(n− k)Πk.

Then, the operators h, L and Λ satisfy the commutator relations:

[h,Λ] = 2Λ ; [h, L] = −2L , and [Λ, L] = h. (10)

The operators L,Λ and h all commute with ∆d, they act on the space ofharmonic forms H∗d(X), and hence on the cohomology ring H∗(X,R). Thisproves:

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Lemma 3.31. Let X be a compact Kahler manifold. The operators L,Λand h induce a representation of sl2(C) on the cohomology of X.

Remark 3.32. These operators also define a representation of sl2 on thespace of differential forms A∗X ; we concentrate on the representation oncohomology because H∗(X,R) is finite dimensional, and we will apply someresults from the theory of finite dimensional representations of sl2(C).

Representations of sl2(C) Recall that sl2(C) is the Lie algebra of SL(2,C),it is the space of 2× 2 matrices of trace 0. A basis of sl2(C) is given by

X =

(0 10 0

), Y =

(0 01 0

)and H =

(1 00 1

),

that satisfy the commutator relations [H,X] = 2X, [H,Y ] = −2Y and[X,Y ] = H.

Definition 3.33. Let V be a finite dimensional C-vector space, and gl(V )the Lie algebra of endomorphisms of V (endowed with the commutator asLie bracket). A Lie algebra representation of sl2(C) on V is a Lie algebrahomomorphism

ρ : sl2(C)→ gl(V ),

i.e. such that for any A,B ∈ sl2(C), ρ([A,B]) = ρ(A)ρ(B)− ρ(B)ρ(A). Thevector space V is a sl2-module.

Any subspace W ⊂ V such that ρ(W ) ⊂ W is an sl2-subrepresentation;for any such subrepresentation, there is a well defined complement W⊥ suchthat V = W ⊕W⊥.

A representation V is irreducible if there is no non-trivial subrepresen-tation.

Any finite dimensional representation V of sl2 is a direct sum of ir-reducible representations. When the representation ρ is fixed, we writeH,X, Y for ρ(H), ρ(X) and ρ(Y ).

Let λ be an eigenvalue for H; λ is called a weight for ρ, and the associatedeigenspace Vλ is a weight space. If λ is aneigenvalue for ρ(H) and v ∈ Vλ,

Hv = λ · v , HXv = (λ+ 2)v and HY v = (λ− 2)v.

Since V is assumed to be finite dimensional, there is a finite number ofweights, and X and Y are nilpotent. A primitive vector v ∈ V is an eigen-value for H such that Xv = 0. The vector subspace of primitive elements isdenoted PV .

60

Proposition 3.34. Let V be an irreducible representation of sl2(C). Ifv ∈ V is a primitive element, V is generated as a vector space by the elementsv, Y v, Y 2v, · · · .

Proof. It is easy to check that the elements Y nv are linearly independent,and that they generate a subrepresentation of sl2(C).

The weights of ρ are easily seen to be integers, the irreducible represen-tation of sl2 are of the form V (n), where V (n) ' SymnC2, where n is theorder of nilpotence of a primitive element v ∈ V for the operator Y . Eachsuch irreducible representation has dimension n+ 1, and

V (n) = V−n ⊕ V−n+2 ⊕ · · · ⊕ Vn−2 ⊕ Vn,

where each Vn−2i is generated by the element Y iv.Any representation V can therefore be written:

V = Vn ⊕ Vn−2 ⊕ · · · ⊕ V−n+2 ⊕ V−n (11)

The Lefschetz decomposition theorem states that there is a direct sum de-composition

V = PV ⊕ Y PV ⊕ Y 2PV ⊕ · · · ⊕ Y nPV, (12)

that is compatible with the decomposition (11). In particular, the weightspaces Vm and V−m are isomorphic, with Xm : V−m → Vm and Y m : Vm →V−m. The primitive elements in Vm are

PV ∩ Vm = ker(Y m+1 : Vm → V−m−2).

The cohomology H∗(X,R) is an sl2-representation, defined by

X 7→ Λ , Y 7→ L , and H 7→ H.

The weight space for the eigenvalue n− k of H is Hk(X,R). The results ofthe theory of representations of sl2(C) imply:

Theorem 3.35 (Hard Lefschetz theorem). Let X be a compact Kahler man-ifold of dimension n. The operator Lk defines an isomorphism

Lk : Hn−k(X,R)→ Hn+k(X,R).

Define

Pn−k = ker(Λ) ∩Hn−k(X,R) = ker(Lk+1 : Hn−k(X,R)→ Hn+k+2(X,R)),

there is a Lefschetz decomposition on the cohomology of X:

Hm(X,R) =⊕

2k≤mLkPm−2k(X).

61

Remark 3.36. Note that since the operator L is of type (1, 1) on forms andsince L and Λ commute with ∆d, the Lefschetz decomposition is compatiblewith the Hodge Decomposition.

Remark 3.37. By definition of the operator L on H∗(X,R), the Lefschetzdecomposition depends on the choice of Kahler class [ω] ∈ H2(X,R); it doesnot depend on the choice of Kahler metric itself (that is, it does not dependon the representative in a fixed class).

Since the class ω is ∂-closed, for all p+ q ≤ n, Ln−p−q defines an isomor-phism

Ln−p−q : Hp,q(X)→ Hn−p,n−q(X). (13)

Exercise 3.38. Show that if X is a compact Kahler manifold, then for allp, q such that k = p + q ≤ n (resp. k = p + q ≥ n ), hp−1,q−1 ≤ hp,q

(resp. hp−1,q−1 ≥ hp,q) and bk ≤ bk+2 (resp. bk ≥ bk+2).

Define an intersection form on the cohomology of X:

Q : Hk(X,R)×H2n−k(X,R)t R

Q(α, β) =

∫Xωn−k ∧ α ∧ β =< Ln−kα, β > .

When k is even, Q is symmetric; when k is odd, Q is alternating. Define aHermitian form on Hk(X,C) by:

Hk(α, β) = ikQ(α, β). (14)

Lemma 3.39. The Lefschetz Decomposition

Hk(X,C) =⊕2r≤k

LrP k−2r(X,C)

is orthogonal for Hk. On each summand LrP k−2r(X,C), the form Hk in-duces (−1)rHk−2r.

Proof. Assume that α = Lrα′ and β = Lsβ′ for some primitive classesα′ ∈ P k−2r(X,C) and β′ ∈ P k−2s(X,C), where s ≥ r. Then:

Hk(α, β) =

∫XLn−kLrα′ ∧ Lsβ′ = (−1)r

∫XLn−k+r+sα′ ∧ β′.

This vanishes if s 6= r because α′ ∈ kerLn−k+2r+1, and coincides with(−1)rHk−2r(α

′, β′) otherwise.

62

Lemma 3.40. The Hodge Decomposition Hk(X,C) =⊕Hp,q(X) is an

orthogonal direct sum for Hk.If Hp,q(X)prim = Hp,q(X) ∩ P k(X,C), the form (−1)k(k−1)/2ip−q−kHk

is positive definite on Hp,q(X)prim.

Proof. If [αp,q] and [βp′,q′ ] are classes in Hk(X,C), Ln−kαp,q ∧ βp′,q′ is of

type (n− k + p+ q′, n− k + p′ + q); this vanishes if (p, q) 6= (p′, q′) becauseH2n(X,C) = Hn,n(X). The Hodge Decomposition on Hk is therefore or-thogonal for Hk. The second part follows from noting that if a cohomologyclass is represented by a harmonic form α, with α primitive, then α is prim-itive at every point x ∈ X, and α is primitive because L and Λ are realoperators.

The Hodge Index Theorem (This was not covered in Lectures)

Exercise 3.41. If ω ∈ Ωp,qX,x is primitive, then

?ω = (−1)k(k+1)/2ip−qLn−kω

(n− k)!.

Theorem 3.42 (Hodge Index Theorem). Let X be a compact Kahler man-ifold of even dimension dimCX = n. The signature of the intersection form

Q(α, β) =

∫Xα ∧ β

on Hn(X,R) is equal to Σp,q(−1)php,q(X).

Proof. The signature of Q is equal to the signature of the Hermitian formH(α, β) =

∫X α ∧ β. Since

Hn(X,C) =⊕2r≤n

LrPn−2r(X,C),⊕

p+q=n−2r≥0

LrHp,q(X)prim

and the sign of H on LrHp,q(X)prim is equal to (−1)p because n is even, sothat:

sgnQ = Σp+q=n−2r(−1)php,q(X)prim.

By Exercise 3.38, this is equal to

sgnQ = Σp+q=n−2r,2r≤n(−1)p(hp,q(X)− hp−1,q−1(X)),

By Poincare Duality, this is equal to:

sgnQ = Σp+q≡n mod 2(−1)php,q(X),

63

and since, by complex conjugation, Σp+q≡1 mod 2(−1)php,q(X) = 0, this is:

sgnQ = Σp,q(−1)php,q(X).

Exercise 3.43. Let X be a compact Kahler surface. Show that the signatureof the intersection form

Q(α, β) =

∫Xα ∧ β

on H2(X,R) ∩H1,1(X) is (1, h1,1 − 1).

4 Hodge Structures

We have seen that the cohomology groups of compact Kahler manifolds areendowed with a very rich structure. If X is a compact Kahler manifold, thedecomposition of differential forms into components of different types de-scends to its Betti or De Rham cohomology groups Hk(X,C) ' Hk

DR(X,C)and yield a direct sum decomposition:

Hk(X,C) =

k⊕p=0

Hp(X,Ωk−pX ) =

k⊕p=0

Hp,k−p(X),

where the summands are related by Hp,k−p(X) = Hk−p,p(X). This decom-position is obtained at level of harmonic forms, that is solutions of par-tial differential equations on X. The Betti cohomology groups also containa lattice, the integral cohomolohy modulo torsion which is the image ofHk(X,Z) ⊗ C → Hk(X,C). Further, cup product with the class of theKahler form induces a Lefschetz decomposition which is compatible withthe Hodge Decomposition, and reflects the existence of a polarisation on thecohomology of X.

These structures are an example of the more general notion of Hodgestructures, which will be well suited to the study of deformations of Kahleror projective manifolds.

4.1 Hodge Structures

Definition 4.1. An integral Hodge structure of weight k is a free abeliangroup of finite type VZ and a decomposition on the complexification:

VC = VZ ⊗Z C =⊕p+q=k

V p,q

64

such that V p,q = V q,p.

Remark 4.2. One can define analogously rational Hodge structures, wherethe lattice VZ is replaced with a finite dimensional Q-vector space VQ.

Remark 4.3. It is sometimes useful to identify Hodge Structures with cer-tain representations of C∗ on VR. For more on this, see Exercise 5, ExampleSheet 6.

Example 4.4. Let X be a compact Kahler manifold, then (HkZ, H

k(X,C) =⊕p+q=kH

p,q(X)) is a Hodge structure of weight k, where HkZ is the torsion

free part of the integral cohomology Hk(X,Z) (i.e. the image of Hk(X,Z)in Hk(X,Q), Hk(X,R) or Hk(X,C)).

Example 4.5. The Tate twist Q(k) is the 1-dimensional weight−2k rationalHodge structure with decomposition:

(Q(k)⊗ C)p,q = (0) if (p, q) 6= (−k,−k) and Q(k)⊗ C)−k,−k = C.

Note that Q(−k) is the Hodge structure on the degree 2k cohomology ofPn for n ≥ k. (This is the Hodge structure associated to the representationz 7→ z−kz−k.)

Example 4.6. Let V be a rational vector space such that VR is endowedwith an almost complex structure J . Then the exterior product

∧k VCalways admit a bidegree decomposition V p,q, which makes it into a Hodgestructure of weight k. For example, when (X,h) is a Hermitian manifold,ΩkX,x has a Hodge structure of weight k for all x ∈ X, but this Hodge

structure does not descends to cohomology unless (X,h) is Kahler.

Example 4.7. The Dolbeault cohomology group Hq(X,ΩpX) of a Kahler

manifold is endowed with a weight p+ q Hodge structure– where the (p, q)-summand is Hq(X,Ωp

X), and the others are zero.(This is the Hodge structureassociated to the representation z 7→ zpz−q.)

Example 4.8. Let (X,ω) be a Kahler manifold, such that [ω] ∈ Hk(X,Z)(resp. [ω] ∈ Hk(X,Q)). The primitive cohomology P k(X,C) is a weight kintegral (resp. rational) Hodge structure.

Definition 4.9. If (VZ, VC =⊕

p+q=k Vp,q) is a weight k Hodge structure,

the associated Hodge Filtration is the decreasing filtration

F •V = F pVC =⊕r≥p

V r,k−r.

65

Remark 4.10. A Hodge structure can alternatively be defined by its inte-gral lattice and the Hodge filtration: the decomposition of VC =

⊕V p,q and

the filtration F •V are equivalent data. By definition of a Hodge Structure,

VC = F pVC ⊕ F k−p+1VC and V p,q = F pVC ∩ F qVC.

In light of Example 4.1, it is natural to ask when Hodge structures onspaces of differential forms descend to Hodge structures on cohomology, ormore generally:

Question 4.11. How do the decompositions on differential forms of degree kand the degree k cohomology classes interact?

Define a filtration on the sheaf of differential forms as follows:

F pAkX = α ∈ Ak(X)| if U ⊂ X,α|U = Σr≥pαr,k−r, with αr,k−r ∈ Ar,k−rX (U),

and denote Πp,k−p : AkX → Ap,k−pX the projection maps.

Remark 4.12. The diagram

Ak(X)d //

Πp,k−p

Ak+1(X)

Πp,k+1−p

Ap,k−p(X)

∂ // Ap,k−p+1(X)

is not in general commutative when p 6= 0.

Exercise 4.13. Show that if X is compact and Kahler, the diagram

C

d // A0(X)d // A1(X)

Π0,1

d // · · ·

OX // A0(X)∂ // A0,1(X)

∂ // · · ·

where the first vertical map is induced by the inclusion of sheaves 0→ C→OX , and the other vertical maps are the projections to the (0, q) component,is commutative. This shows that the map on cohomology Hk(X,C) →Hk(X,OX) induced by the inclusion of sheaves C ⊂ OX coincides with thatinduced by the bidegree decomposition.

Proposition 4.14. If X is a compact Kahler manifold, then:

F pHk(X,C) =ker(d : F pAk(X)→ F pAk+1(X))

im(d : F pAk−1(X)→ F pAk(X)).

66

Proof. There is a natural map f from ker(d : F pAk(X) → F pAk+1(X)) toHk(X,C) which sends a closed form α to its De Rham cohomology class.The image of f is contained in F pHk(X,C).

By Corollary 3.11, every class in F pHk(X,C) has a unique harmonicrepresentative. Let [γ] ∈ F pHk(X,C) be a cohomology class with γ har-monic. Since γ ∈ ker(d : F pAk(X)→ F pAk+1(X)), the map f is surjective.We want to show that if α ∈ ker f , α ∈ im(d : F pAk−1(X) → F pAk(X)).By induction, first assume that α ∈ F pApX is such that dα = 0 and [α] = 0.

The form α ∈ Ap,0X , and by Lemma 3.25, α = ∂∂β with β ∈ Ap−1,−1X . This

shows that α = 0 because Ap−1,−1X = 0.

Assume that α ∈ F pAkX is a closed form such that [α] = 0. Writeα = αp,q + α′ for α′ ∈ F p+1AkX . Write αp,q = βp,q + ∆∂ε

p,q, where βp,q is∂ (and d)-harmonic. Since αp,q is d-exact, by Lemma 3.25, the Dolbeaultcohomology class of αp,q is zero and βp,q = 0, so that αp,q = ∆∂ε

p,q. Since

∂αp,q = 0, ∆∂εp,q = ∂∂

∗εp,q. The form ∂

∗εp,q is a section of Ap,q−1

X , so that

δ∂∗εp,q ∈ Ap+1,q−1

X , so that the form α− d(∂∗εp,q) ∈ F p+1AkX , and this form

is d-closed and d-exact. By induction, α − d(∂∗εp,q) ∈ im(d : F p+1Ak−1

X →F p+1AkX), and hence α ∈ im(d : F pAk−1(X)→ F pAk(X)).

An immediate consequence of this proposition is:

Corollary 4.15. For all p ∈ N, Hp,0(X) ' Γ(X,ΩpX).

Polarisations

Definition 4.16. An integral Polarised Hodge Structure of weight k is anintegral Hodge Structure of weight k (VZ, VC =

⊕V p,q) endowed with an

intersection formQ : VZ × VZ → Z

such that

1. Q is symmetric if k is even and alternating otherwise,

2. For all non-zero α ∈ V p,q, ip−q−k(−1)k(k−1)/2H(α) > 0, whereH(α, β) =ikQ(α, β).

Remark 4.17. In terms of representations of C∗ on VR (see Ex.5, ES 6),these conditions can be reformulated as follows: if

Q : VZ × VZ → Z,

setting (ρ(z)α, ρ(z)β) = zkzk(α, β), the form (·, ρ(i)·) is symmetric and def-inite positive.

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Example 4.18. Let (X,ω) be a Kahler manifold with [ω] ∈ H2(X,Z).Then the primitive cohomology P k(X,C) is a polarised Hodge structure ofweight k.

Remark 4.19. As above, taking Q : VQ × VQ → Q, one defines a rationalpolarised Hodge structure. If the Kahler class is rational, the primitivecohomology P k(X,C) is a rational polarised Hodge structure of weight k.

Definition 4.20. A polarised manifold (X,ω) is a compact Kahler manifoldwhose Kahler class [ω] is integral, i.e. [ω] ∈ H2(X,Z).

The significance of this notion is illustrated in the following theorem–which we have already seen (Theorem 2.33)– but recall here.

Theorem 4.21 (Kodaira Embedding Theorem). Let X be a compact com-plex manifold. The following are equivalent:

1. X is projective, i.e. there exists a holomorphic embedding X → PN forsome N ∈ N.

2. There exists an integral Kahler form ω on X; [ω] ∈ H2(X,R) ∩H1,1(X).

3. There exists a positive holomorphic line bundle L→ X.

Corollary 4.22. Let X be a compact Kahler manifold. If H2(X,OX) = (0),X is projective.

Proof. The compact manifold X is Kahler, hence H2(X,OX) = H0,2(X) =H2,0(X), and H2(X,C) = H2(X,Q) ⊗ C = H1,1(X). Since H2(X,C) 'H2(X), there exists a basis α1, · · · , αr of H2(X,Q) consisting of harmonicforms. The Kahler form ω itself is harmonic (see Ex.7, ES 5), and real, soit is a linear combination

ω = Σrj=1λjαj ,

where λj ∈ R for all j. By Ex. 6, ES 5, KX ⊂ H1,1(X), the set of Kahlerclasses on X, is an open convex cone, so that if qj ∈ Q are such that|qj − λj | ≤ ε for some 0 < ε << 1, and the class of

ω′ = Σrj=1qjαj

is Kahler and lies in H2(X,Q). An appropriate multiple of ω′ is then anintegral Kahler form, and (X, [nω′]) is a polarised manifold. The resultfollows by the Kodaira Embedding Theorem.

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4.2 Weight 1 Hodge Structures and complex tori

We first establish some results on the cohomology of complex tori.Let VR be a 2n-dimensional R-vector space, and let Z2n ' Λ ⊂ VR be

a full lattice. The complexified vector space VC = VR ⊗ C has a naturaldecomposition V ⊕ V . The image of Λ by the inclusion map VR → VC is afull lattice Z2n in V ' Cn, and T = V/Λ is a complex torus.

Exercise 4.23. Every complex torus T = V/Λ is of this form (set VR = Λ⊗R).

The complex torus T inherits a group structure from Λ as follows. Let:

π : V → T

be the natural projection map. If a ∈ T , let va be a point in V with π(va) =a. The translation automorphism τa : T → T is defined by τa(x) = π(v+va),where v ∈ V is such that π(v) = x.

Exercise 4.24. Check that the automorphism τa is well defined.

Near a point P ∈ T , a complex basis of V yields local holomorphiccoordinates of T , and there is a natural identification

T 1,0T,p ' V.

A Hermitian inner product on V yields a Kahler metric on T , which isinvariant under translations τa for a ∈ T . We assume that T is endowedwith such a metric.

Denote I∗(T ) = α ∈ A∗T |τ∗a (α) = α for all a ∈ T

Lemma 4.25. The space of harmonic forms H∗(T ) coincides with I∗(T ).If z1, · · · , zn are local holomorphic coordinates on T induced by a basis of V ,H∗(T ) = C · dzI ∧ dzJI,J , for I, J ⊂ 1, · · ·n.

Proof. Let a ∈ T , since τ∗a : A∗T → A∗T preserves the metric, τ∗a : H∗(T ) →H∗(T ). The space H∗(T ) is isomorphic to H∗(T,C) because T is Kahler andτ∗a is homotopic to the identity for all a ∈ T , it follows that H∗(T ) ⊂ I∗(T ).

Conversely, if α ∈ I∗(T ), α is entirely determined by its value at anypoint P ∈ T , and since TT,P,C = V ⊕ V , I∗(T ) '

∧∗(V ∗)⊗∧∗(V ∗), whereV ∗ is the dual of V .

Since T ' (S1)2n, bk(T,C) = dimHk(T ) = dim Ik(T ), and this con-cludes the proof.

The homology of T is well known–any loop γ with base point 0 ∈ T liftsto a path γ in V ' Cn, the universal covering space of T , with end pointλ ∈ Λ, so that H1(T,Z) ' Λ. We have shown the following:

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Lemma 4.26. If T is a complex torus,

T ' H1,0(T )∗/H1(T,Z).

A complex torus T is entirely determined by the weight 1 Hodge Structure(H1(T,Z), H1(T,C)).

There are two distinguished bases of Hk(T,Z):

• A Z-basis λ1, · · · , λ2n of H1(T,Z) yields a dual basis x1, · · · , x2nfor V as a R-vector space. Denote dx1, · · · , dxn the associated 1-forms,then, by definition, ∫

λi

dxj = δi,j ,

and Hk(T,Z) is generated by forms dxI , for |I| = k;

• Lemma 4.25 shows that Hk(T,Z) is generated by forms dzI∧dzJ , with|I|+ |J | = k.

The first basis reflects the integral structure, while the second reflects thecomplex structure. Conditions can be expressed for T to be polarised– orequivalently, projective–in terms of transition matrices between these bases(see [GH78]).

Let now X be a compact Kahler manifold, by Theorem 3.18, there is adecomposition:

H1(X,C) = H1,0(X)⊕H0,1(X) and H1,0(X) ' H0,1(X).

Assume that H i(X,Z) is torsion free for all i > 0–if this is not the case,we replace H i(X,Z) by its image in H i(X,R)). The long exact sequenceassociated to the exponential exact sequence

0→ Z 2iπ·→ OXexp→ O∗X → 0 (15)

shows that H1(X,Z) → H1(X,OX). From the computations on singularcohomology, we have seen that:

H1(X,Z)i1→ H1(X,R)

i2→ H1(X,R)⊗ C = H1(X,C)

pr→ H0,1(X),

and we have seen that pr i2 : H1(X,R) → H0,1(X) is an isomorphism ofR-vector spaces. The image of H1(X,Z) in H0,1(X) is a lattice of rankb1(X).

We have proved:

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Definition-Lemma 4.27. Let X be a compact Kahler manifold, the Picardvariety of X is Pic0(X) = H0,1(X)/H1(X,Z); Pic0(X) is a complex torus.

The Picard variety Pic0(X) is Kahler, the long exact sequence in coho-mology associated to (15) shows that Pic0(X) fits in an exact sequence:

0→ Pic0(X)→ Pic(X)c1→ im c1 → 0. (16)

By Theorem 3.18, the subgroup im c1 ⊂ H2(X,Z) ∩ H1,1(X) is called theNeron-Severi group of X and denoted NS(X).

By (16), Pic0(X) ' L ∈ PicX : c1(L) = 0; Pic0(X) parametrisesnumerically trivial line bundles.

Remark 4.28. The Lefschetz theorem on (1, 1)-classes (see Exercise 2, ES4) shows that NS(X) = H2(X,Z) ∩H1,1(X).

The Picard group Pic(X) is therefore made of a discrete part, NS(X)and a continuous part, Pic0(X), that is a complex torus.

Let α ∈ NS(X) and Lα ∈ Pic(X) be a line bundle such that c1(Lα) = α.Denote

Picα(X) = L ∈ Pic(X) : c1(L) = α,

then, by Remark 1.36, the map L 7→ L ⊗ L−1α defines an isomorphism

Pic0(X) ' Picα(X). This isomorphism depends on the choice of a dis-tinguished line bundle in Picα(X), it is not canonical.

If X is a compact Kahler manifold and H2(X,Z) is torsion free, Pic(X)is fibered by complex tori of dimension b1(X) over a (discrete) subgroup ofH2(X,Z).

There is another complex torus canonically associated to a compactKahler manifold X, the Albanese variety. Indeed, by Serre duality, H1(X,C)is dual to H2n−1(X,C), with

H1,0(X) ' Hn−1,n(X)∗ and H0,1(X) ' Hn,n−1(X)∗;

while Poincare duality statesH2n−1(X,Z) ' H1(X,Z)∗. We identifyH1(X,Z)∗

with its image in H2n−1(X,Z).

As in the case of the Picard variety, the image of H2n−1(X,Z) by thecomposition of maps

H2n−1(X,Z)i1→ H2n−1(X,R)

i2→ H2n−1(X,R)⊗C = H2n−1(X,C)

pr→ Hn−1,n(X)

is a full lattice.

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Definition 4.29. The Albanese variety of X is the complex torus

Alb(X) = Hn−1,n(X)/H2n−1(X,Z).

Remark 4.30. We may identifyH1(X,Z) with its image by the mapH1(X,Z)→H0(X,ΩX)∗ defined by:

[γ] 7→ Iγ where Iγ(α) =

∫γα.

Check that when X is Kahler, this map is well defined (i.e. that Iγ dependsonly on the class of the loop [γ]). The Albanese and Picard varieties aredual tori :

Alb(X) ' H0(X,ΩX)∗/(H1(X,Z)).

We now show that the Albanese variety is universal from the point ofview of maps from X to complex tori. Fix a base point x0 ∈ X and definethe Albanese map:

alb : X → Alb(X)

x 7→ (α→∫ x

x0

α).

Exercise 4.31. Check that the Albanese map is well defined and determinethe effect of a change of base point in its definition.

Proposition 4.32. The Albanese map is holomorphic and the pullbackof forms induces an isomorphism H0(X,ΩX) ' H0(Alb(X),ΩAlb(X)). Iff : X → Y is a holomorphic map between Kahler manifolds, f induces acommutative diagram

Xf //

alb

Y

alb

Alb(X) // Alb(Y )

where the basepoint of Alb(Y ) is f(x0), and the map Alb(X) → Alb(Y ) isinduced by the pullback of forms.

Sketch proof. Holomorphicity is a local question: for any 1-form α, in theneighbourhood of x1 ∈ X, the holomorphicity of alb(x) =

∫ x1x0α +

∫ xx1α is

equivalent to the holomorphicity of x 7→∫ x

0 α on a polydisc D ⊂ Cn, andthis is clear.

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Since f is holomorphic, the pullback of forms preserves the type of forms,and hence defines a map H1,0(Y )→ H1,0(X). The existence of the diagramthen follows from the definition of pullbacks because for any path γ from x0

to x in X,∫ xx0f∗α =

∫ f(x)f(x0) α.

Exercise 4.33. Show that Alb(Alb(X)) = Alb(X).

Exercise 4.34. Show that when dimX = 1, Alb(X) ' Pic0(X) (this torusis then called the Jacobian of X).

Exercise 4.35. Show that Pic0(Alb(X)) ' Pic0(X), and that Alb(Pic0(X)) 'Alb(X).

Exercise 4.36. Show that if X is a compact Kahler manifold, with b1(X) = 0,then every holomorphic map X → T , where T is a complex torus is constant,and that PicX ' NS(X).

Exercise 4.36 can be generalised as follows:

Corollary 4.37. Let f : X → T be a holomorphic map from X to a complextorus, then f decomposes as X → Alb(X)→ T .

Proof. This is a direct consequence of the diagram of Proposition 4.32, be-cause T = Alb(T ).

It is natural to ask under which conditions these complex tori naturallyassociated to X are projective.

Definition 4.38. A complex torus is an abelian variety if it is projective.

By the Kodaira Embedding Theorem (2.33), since a complex torus T isKahler, this is equivalent to asking whether there exists an integral Kahlerform on T .

Remark 4.39. It is possible to write precise conditions for a complex torusto be projective; these (the Riemann conditions) are formulated in termsof the transition matrices between the bases of Hk(T,Z) associated to theintegral and complex structures mentioned above.

Lemma 4.40. If X is projective, Pic0(X) is an abelian variety.

Proof. Since X is projective, H1(X,C) = P 1(X,C) is an integral PolarisedHodge Structure, i.e. there exists an alternating form

Q : H1(X,Z)×H1(X,Z)→ Z,

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with Q(α, β) = (Ln−1α, β)L2 , where L is the Lefschetz operator and n =dimX.

The form Q takes integral values on H1(X,Z), H1,0(X) and H0,1(X)are isotropic subspaces for Q, and the Hermitian form H(α, β) = iQ(α, β)is positive definite on H1,0(X). By the description of differential forms ona torus given above, for any P ∈ Pic0(X), we have:

TPic0(X),P ' H1(Pic0(X),Z)⊗R ' H1(X,Z)∗⊗R, and H1,0(Pic0(X)) ' H0,1(X)

so that Q ∈∧2(H1(X,Z))∗ defines an element of

∧2(H1(Pic0(X),Z)∗)∗ andhence of H2(Pic0(X),Z)∗.

We denote [Ω] the class of the element of H2(Pic0(X),R) obtained byextending this element by R-linearity. Note that by construction, [Ω] ∈H2(Pic0(X),Z) is an integral class. The form Ω is of type (1, 1) because Qvanishes on

∧2 T 1,0

Pic0(X)' Λ2H0,1(X).

Last, Ω is positive if u ∈ TPic0(X),R, and u = 2<(u1) for u1 ∈ H0,1(X) 'T 1,0

Pic0(X),C, we have:

Ω(u, Iu) = Ω(u1 + u1,−iu1 + iu1) = 2iQ(u1, u1) > 0

because H is definite positive on H1,0(X). The class [Ω] ∈ H2(Pic0(X),R)is an integral Kahler class and the result follows by Theorem 2.33.

4.3 Functoriality

Let (VZ, F pVCp) and (WZ, F qWCq) be integral Hodge structures of weightsn and m = n+ 2r respectively, for r ∈ Z.

Definition 4.41. A morphism of Hodge structures is a group homomor-phism ϕ : VZ → WZ whose extension by C-linearity ϕC : VC → WC satisfiesϕ(F pVC) ⊂ F p+r(WC) for all p ∈ N. The morphism ϕ is said to be of type(r, r).

Remark 4.42. Note that the condition ϕ(F pVC) ⊂ F p+rWC is equivalentto requiring that ϕ(V p,q) ⊂W p+r,q+r for all p, q ≥ 0.

Assume that ϕ : (VZ, F pVCp)→ (WZ, F qWCq) is a morphism of type(r, r). It is easy to check that ϕ is strict, that is: imϕ∩W p+r,q+r = ϕ(V p,q),and hence induces a Hodge structure of weight m on imϕ. Similarly, ϕinduces a natural Hodge structure of weight n on kerϕ, and a natural Hodgestructure of weight m on Cokerϕ.

There are several “natural” morphisms of Hodge structures from thepoint of view of geometry.

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Pullbacks Let ϕ : X → Y be a holomorphic map.

First, ϕ is a continuous map of topological spaces, so that there is apullback map

ϕ∗ : Hksing(Y,Z)→ Hk

sing(X,Z) (17)

induced by the morphism of complexes given by ϕ∗(α)(ψ) = α(ϕ ψ) forany singular cochain α on Y , and singular chain ψ on X.

Second, since ϕ is a differentiable map, there is a pullback map of dif-ferential forms ϕ∗ : A•Y → A•X such that ϕ∗ dY = dX ϕ∗, so that ϕ∗ liftsto the cohomology groups. The map ϕ∗ between the De Rham complexesϕ∗ : A•Y → ϕ∗A•X naturally extends the map ϕ∗ : ZY → ϕ∗ZX inducing (17),and defines a map

ϕ∗ : Hk(Y,C)→ Hk(X,C).

Since ϕ is holomorphic, ϕ∗ preserves the type of forms, and ϕ∗ is amorphism of Hodge structures of type (0, 0). Note that by definition, ϕ∗ iscompatible with cup products–that is, ϕ∗(α ∪ β) = ϕ∗α ∪ ϕ∗β.

Lemma 4.43. Let ϕ : X → Y be a holomorphic and surjective map ofcompact Kahler manifolds. Then, ϕ∗ : Hk(Y, Z)→ Hk(X,Z) is injective.

Remark 4.44. Here we assume implicitly that the cohomology groupsHk(X,Z) and Hk(Y,Z) are torsion free; if this is not the case, the statementof the Lemma concerns the torsion free part of the integral cohomology.

Proof. By Remark 4.44, it is enough to prove the result for the cohomologywith real or complex coefficients. Let n = dimY and n + r = dimX. Wefirst prove that H2n(Y,R) → H2n(X,R) is injective. To this end, considera nonzero class [α] ∈ H2n(Y,R) ' R, we prove that ϕ∗α 6= 0. We mayassume that [α] ∈ H2n(Y,C) is represented by a positively oriented form α.Let ω be a Kahler form on X, then ϕ∗α ∧ ωr is positive or 0 at every pointof X. At every point where dϕ is a submersion, ϕ∗α ∧ ωr 6= 0; this is thecase on an open set U ⊂ X and hence

∫X ϕ

∗α ∧ ωr > 0 in H2n+2r(X,R).It follows that ϕ∗[α] 6= 0 in H2n(X,R). Consider an arbitrary nonzero class[α] ∈ Hk(Y,R). By Poincare Duality, there is a class [β] ∈ H2n−k(Y,R) suchthat [α] ∪ [β] = [η] is a positive class in H2n(Y,R). We have noted that ϕ∗

is compatible with cup products, and the result follows.

Remark 4.45. Note that the assumption that X and Y are Kahler is onlyused when the dimensions of X and Y are different.

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Pushforwards Let ϕ : X → Y be a holomorphic map between complexmanifolds of dimensions dimX = n and dimY = n + r = m. Consider themap on singular homology dual to (17):

H2n−k(X,Z)→ H2n−k(Y,Z) = H2m−2r−k(Y,Z). (18)

The pushforward map

ϕ∗ : Hk(X,Z)→ Hk+2r(Y,Z)

is the map that is Poincare dual to (18). If ψ : ∆ → X is a singular chain,ϕ∗(ψ) is the singular chain ϕ ψ : ∆→ Y .

The induced morphism of Hodge structures ϕ∗ is of type (r, r). We needto check that if a differential form α is of type (p, q) on X, ϕ∗α is of type(p+ r, q + r). This follows immediately from the fact

Hk,l(Y ) = (⊕

p′+q′=2m−k−l,(p′,q′)6=(p,q)

Hp′,q′(Y ))⊥,

where the “orthogonality” is with respect to Poincare duality.

Hodge structure of a blowup Let σ : XY → X be the blowup of aKahler manifold X along a submanifold Y ; denote E the exceptional divisorof σ and recall that E ' P(NY/X). If Y has codimension r, E is a Pr−1-bundle over Y . We want to determine the relations between the cohomologygroups Hk(XY ,Z), Hk(X,Z) and Hk(Y,Z).

Intuitively, since E → Y is a projective bundle, we expect that Hk(E,Z)and Hk(Y,Z). The following theorem, which we admit, formalizes this in-tuition.

Theorem 4.46 (Leray-Hirsch Theorem). Let π : F → Z be a fibration overa locally contractible base Z. Suppose that H•(Fz,Z) is torsion free forall z ∈ Z. Assume there are classes α1, · · · , αN ∈ H•(F,Z) such that thesubgroup A =< α1, · · · , αN > of H•(F,Z) is isomorphic by restriction toHk(Fz,Z) for all z ∈ Z, then

Hk(F,Z) ' A⊗Z H•(Z,Z),

where the isomorphism is obtained by restriction and cup product. Moreprecisely, this means that any cohomology class in Hk(F,Z) is of the formh ∪ α, where h is a class of degree j in A, and α ∈ Hk−j(Z,Z).

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Remark 4.47. You may compare this result with the construction of theHodge bundles H in the next section.

Lemma 4.48. The cohomology H•(E,Z) is a free module over H•(Y,Z)with basis 1, h, · · · , hr−1, where h = c1(OE(1)).

Proof. This is a direct consequence of the Leray-Hirsch theorem. Indeed,Hk(Ez,Z) is torsion free for all z ∈ Z and we have seen that c1(OE(1))|Ex =c1(OP(Ex)(1)) for all x ∈ Y , so that H•(E,Z) =< 1, h, · · · , hr−1 >, with

h ∈ H2(E,Z). Leray-Hirsch then implies that any class in Hk(E,Z) is ofthe form hi ∪ α for α ∈ Hk−2i(Z,Z).

Theorem 4.49. Let X be a Kahler manifold, Y ⊂ X a submanifold ofcodimension r, and let XY

σ→ X be the blowup of X along Y ; then

Hk(X,Z)⊕ (

r−2⊕i=0

Hk−2i−2(Y,Z))σ∗+Σj∗hiσ∗|E−→ Hk(XY ,Z) (19)

is an isomorphism of Hodge Structures. In (19), hi denotes the cup productwith the class hi = (c1(OE(1)))i, and is a morphism of Hodge structures oftype (i+ 1, i+ 1), j : Y → X is the inclusion and σ|E is the restriction of σto E = Excσ.

Proof. By definition of the blowup map, XrY ' XY rE ' U . The blowupthus defines an isomorphism of pairs σ : (XY , U)

'→ (X,U) which induces adiagram:

Hk−1(U)

// Hk−1(X,U)

σ∗X,U

jY ∗ // Hk(X)

σ∗X

// Hk(U)

Hk−1(U) // Hk−1(XY , U)

j∗ // Hk(XY ) // Hk(U)

(20)

where the first and last vertical arrows are isomorphisms. Here, we omit thecoefficient ring Z from the notation of the cohomology groups. The Excisionand Thom isomorphism Theorems in Algebraic Topology imply that:

Hk−1(X,U) ' Hk−2r(Z) and Hk−1(XY , U) ' Hk−2(E).

Lemma 4.43 shows that σ∗ and σ∗X,U are injective and Lemma 4.48 showsthat σ∗X,U coincides with the map:

α : Hk−2r(Z)→r−1⊕i=0

hi(σ∗|EHk−2i−2(Z,Z)) ' Hk−2(E).

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The commutativity of the diagram (20) shows that

(σ∗, j∗) : Hk(X)⊕Hk−2(E)→ Hk(XY )

is surjective. Also, injectivity of σ∗X,U : Hk−1(X,U)→ Hk−1(XY , U) impliesthat

ker(σ∗, j∗) = im((jZ∗,−σ∗X,U ) : Hk(Z)→ Hk(X)⊕Hk−2(E).

The theorem then follows from the description of the cohomology of E inLemma 4.48.

5 Deformations, families and the Kodaira-Spencermap

We have seen in the previous sections that the cohomology of a compactKahler manifoldX is endowed with a Hodge Structure (Hk(X,Z), Hk(X,C) =⊕Hp,q(X)), which encodes much geometric information. This Hodge Struc-ture does not depend on the Kahler metric itself, but on the complex struc-ture on X. In a sense, the Hodge Decomposition on the cohomology ofX means that, under the Kahler hypothesis, the decomposition induced bythe complex structure on the exterior algebra of differential forms descendsto cohomology. It is natural to try and understand the dependency of theHodge Structure on the complex structure. There are several ways to for-mulate this problem.

First, we could ask how the Hodge structure on a Kahler manifold Xchanges when the underlying differentiable manifold X is fixed while thecomplex structure varies. Recall that the complex structure on X can bedefined in the following two equivalent ways.

1. X is endowed with a complex atlas, i.e. an open cover X = ∪Ui anda collection of holomorphic charts ϕi : Ui → Cn. Two such structuresUi, ϕi and Vj , ψj are equivalent if there is a map f : X → X suchthat ϕi f ψ−1

j is holomorphic for all i, j.

2. X is endowed with a complex structure, i.e. an endomorphism I : TX,R →TX,R, with I2 = Id that is integrable. Two such structures (X, I) and(X, I ′) are equivalent if there is a diffeomorphism F : X → X suchthat dF I = I ′ dF .

Alternatively, we could ask how the Hodge structure varies in a familyof complex manifolds. In other words, if X → B is a proper submersive

78

map between complex manifolds, such that Xt is Kahler for all t ∈ B, weask whether the Hodge structures on the cohomology of Xt and on Xt′ fort 6= t′ can be related.

We will see that these two formulations are in fact equivalent; a familyof compact complex manifolds is locally a family of complex structures ona fixed differentiable manifold.

5.1 Small deformations of complex manifolds

Definition 5.1. A family of complex manifolds is a proper holomorphic andsubmersive map π : X → B between connected complex manifolds.

Remark 5.2. The assumption that π is proper and holomorphic impliesthat Xt is a compact complex submanifold of X for all t ∈ B. Since πis submersive, π has maximal rank everywhere, X is locally diffeomorphicto a product, i.e. for all t ∈ B, there is a neighbourhood V of t and adiffeomorphism T : X|π−1(V ) ' U × V , where U ⊂ Cn is an open set andpr2 T = ϕ.

Theorem 5.3 (Ehresmann Theorem). Let π : X → B be a proper submer-sive map between differentiable manifolds. If B is contractible, there is a

diffeomorphism X T' X0 ×B, where 0 ∈ B is a basepoint and X0 = π−1(0).

Proof. We use the following property of real differentiable manifolds (Tubu-lar Neighbourhood Theorem): there is a neighbourhood U of X0 in Xthat is diffeomorphic to a neighbourhood V of X0 in the normal bundleNX0/X , and a differentiable retraction T : U → X0. The differentiable mapF = (T, π) : U → X0 ×B has invertible differential dF along X0, and hencedefines an embedding in a neighbourhood of X0. There is an open set V ⊂ Bsuch that U ⊂ π−1(V ) , and F = (T, π) : π−1(V )→ X0× V is a diffeomorp-shim.

By assumption, B is contractible, hence there is a vector field v ∈H0(B, TB) with flow Φ that exists for all t such that im Φt ⊂ V for t >> 0.Let k = dimRB. There is a differentiable chart of ∪Ui, ϕ : Ui → Rm+kthat trivialises π, i.e. such that π ϕ−1

i is the projection to the last k coordi-nates. The vector field v lifts to each Ui, and by using a partition of unity,there is a lift v of v to X. The associated flow Ψ exists for all t because πis proper, and by definition of v, π Φ = Ψ π; Ψt defines a diffeomorphismbetween X and π−1(V ) that is compatible with π.

Remark 5.4. Note that the map T is a diffeomorphism and not biholomor-phic. The fibres of T are not in general biholomorphic, but the diffeomor-

79

phisms Tt : Xt → X0 allow us to compare the complex structures on fibresof π.

Proposition 5.5. Let π : X → B be a family of complex manifolds and 0 ∈B. Possibly shrinking B to a neighbourhood of 0 ∈ B, there is a differentiabletrivialisation T = (T0, π) : X → X0 ×B such that for all x ∈ X0, T−1

0 (x) isa complex manifold. Note that the submanifold T−1

0 (x) is diffeomorphic toB for every x ∈ X0.

Sketch Proof. As in the proof of Theorem 5.3, there is a diffeomorphismψ : V → U between a neighbourhood of X0 in NX0/X and a neighbourhoodof X0 in X . We may assume that the restriction of ψX0 to the 0-sectionX0 ⊂ NX0/X is the inclusion X0 → U , that dψX0 induces a canonical iso-morpshim NX0/X ' NX0/U , and that ψ is real analytic in the neighbourhoodof 0 in each fibre of V → X0 (this is part of the statement of the tubularneighbourhood theorem).

The map ψ associates to every (x, u) ∈ V ⊂ NX0/X the end point of thegeodesic γ defined by γ(0) = x, γ(0) = u. For all x ∈ X0, ψx : π−1(x) =Ux → X is real analytic –we admit this fact, it is a consequence of the tubularneighbourhood theorem–and gives expansions of holomorphic coordinatesz1, · · · , zn on X in terms of real analytic coordinates x1, · · · , xm on U . Defineψhx as the holomorphic part of ψx. The definition of ψhx is independent ofthe choice of coordinates on X .

The differentiable map ψ : U → X is holomorphic on the fibres of π.The restriction to the 0-section of d(π ψ)|X0

= d(π ψh)|X0is C-linear

and surjective. Denote π′ : U → X0 the map induced by ψ, then T =(π′ ψh, π) : X → X0 × B is a diffeomorpshim such that for all x ∈ X0,T−1

0 (x) is a complex manifold.

Remark 5.6. By Theorem 5.3, in a neighbourhood of 0 ∈ B, the fibres of πare diffeomorphic to X0. Proposition 5.5 shows that the family of complexstructures on X0 parametrised by B varies holomorphically with t ∈ B.

We now try and classify deformations of complex manifolds. Let π : X →B be a family of complex manifolds and X0 = π−1(0) its central fibre. Thedifferential dπ : TX → π∗TB is a surjective morphism of holomorphic vectorbundles and we define the relative tangent bundle TX/B as the kernel of dπ.Restricting dπ to X0 yields an exact sequence:

0→ TX0 → (TX )|X0→ (π∗TB)|X0

→ 0, (21)

80

where (π∗TB)|X0' π∗TB,0×X0. Hence, (21) defines an extension of TX0 by

(π∗TB)|X0; this extension is characterised by the map (see ES3, ex 7):

ρ : H0(X0, (π∗TB)|X0

)→ H1(X0, TX0),

the connecting map in the long exact sequence associated to (21). Note thatsince (π∗TB)|X0

is trivial,

ρ : TB,0 → H1(X0, TX0).

Definition 5.7. The map ρ : TB,0 → H1(X0, TX0) is the Kodaira-Spencermap of the family X → B at 0 ∈ B.

There are several ways to think about the Kodaira-Spencer map. LetT = (T0, π) be the map of Proposition 5.5. For t ∈ B, Xt and X0 arediffeomorphic, denote Tt the induced diffeomorphism.

Using the diffeomorphism Tt, the complex structure It ∈ End(TXt,R) onXt defines a complex structure on X0. We consider the map

t ∈ B → It ∈ End(TX0,R).

Giving a complex structure I on X0 is equivalent to giving a direct sumdecomposition of TX0,R into eigenspaces for I. In particular, giving It is

equivalent to specifying a subspace (T 1,0X0,x

)t ⊂ TX0,x,C that varies differen-tiably with x ∈ X0. We show that the map above corresponds to a map:

t 7→ αt ∈ A0,1(T 1,0X0

)

where αt(x) ∈ Ω0,1X0,x⊗ T 1,0

X0,x, is the map

T 0,1X0,x

(T−1t )∗

' (T 0,1X0,x

)tpr2→ T 1,0

X0,x.

Here pr2 denotes the projection onto the second factor of the decompositionTX0,x,C = T 0,1

X0,x⊕ T 1,0

X0,x.

Conversely, if αt ∈ Ω0,1X0,x⊗ T 1,0

X0,x, αt defines a complex structure on X0

(and hence on Xt, using the diffeomorphism Tt), for which the vectors oftype (0, 1) are of the form u− αt(u) for all u ∈ T 0,1

X0,x.

Proposition 5.5 implies that t 7→ αt is holomorphic; also, by definition,α0 = 0.

81

Proposition 5.8. The map

TB,0 → A0,1(TX)

u 7→ du(αt)

takes values in σ ∈ A0,1(TX) : ∂σ = 0. If u ∈ TB,0, the Dolbeault cohomol-ogy class of du(αt) in H1(TX) is ρ(u), the image of u by the Kodaira-Spencermap.

Proof. Let T−1 : X0×B'→ X be the inverse of the diffeomorphism of Propo-

sition 5.5. We have seen that for all x ∈ X0, T−1(x×B) is a complex subman-ifold, which defines a complex sub-vector bundle T−1

∗ (TB) = π∗(TB) ⊂ T 1,0X ,.

To this subbundle, is associated a differentiable map σ : π∗TB → TX , andhence a differentiable splitting of the exact sequence

0→ TX/B → TX → π∗TB → 0.

When restricted to X0, this defines a differentiable splitting of

0→ TX0 → (TX )|X0→ π∗TB,0 → 0.

By definition, the Dolbeault cohomology class ρ(u) is the class of ∂(σ(u))for u ∈ TB,0.

We have to prove that ∂σ(u) = du(αt) ∈ A0,1(TX). Since this is a lo-cal statement, we will introduce coordinate systems. Let t1, · · · , tk be localholomorphic coordinates for B and z1, · · · , zn, t1, · · · , tk be local holomor-phic coordinates for X (there exist such coordinates for X because π isholomorphic and submersive). The diffeomorphism X → X0 is

T0 : (z1, · · · , zn, t1, · · · , tk) 7→ (f1, · · · , fn),

where fi1≤i≤n are differentiable functions that are holomorphic in thevariables ti. Vector fields of type (0, 1) for It are of the form:

(T0)∗(∂

∂zi) = Σj

∂fj∂zi

(∂

∂zj) + Σj

∂fj∂zi

(∂

∂zj). (22)

We have seen that vectors of type (0, 1) for It are of the form u−αt(u) for ua vector field of type (0, 1) for I0, i.e for u a C∞ combination of the vectorfields ∂

∂zk. From (22), we have:

αt(Σj∂fj∂zi

(∂

∂zj)) = −Σj

∂fj∂zi

(∂

∂zj)

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Since α0 = 0 and f(z, 0) = z, at the first order in t, this shows that

αt(∂

∂zi) = −Σj

∂fj∂zi

(∂

∂zj)

Differentiate this with respect to tk to obtain:

∂αt∂tk|t=0(

∂

∂zi) = − ∂

∂ziΣ∂fj∂tk

(∂

∂zj).

Now consider σ( ∂∂tk

). By definition of σ, the vector field σ( ∂∂tk

) is of type

(1, 0) and π∗(σ( ∂∂tk

)) = ∂∂tk

on B and T0∗(σ( ∂∂tk

)) = 0. Since T0∗(∂∂tk

) =

Σj∂fj∂tk

∂∂zj

, we obtain:

σ(∂

∂tk) =

∂

∂tk− Σj

∂fj∂tk

∂

∂zj,

and the result follows.

There is another way to think about the action of the Kodaira-Spencermap, which is slightly more algebraic.

Define the thick point at 0 as the complex manifold Bε whose under-lying topological space is the point 0 ∈ B, endowed with the sheaf ofholomorphic functions OBε = OB/(m0)2, where m0 is the maximal ideal of0 ∈ B.

The first order neighbourhood Xε of X is defined by the commutativediagram:

Xε//

X

Bε // B,

where the bottom map is the scheme theoretic inclusion. In other words, Xε

has the same underlying topological space as X, and its sheaf of holomorphicfunctions is OXε = π∗zi mod m2

0; zi a normal direction to X0 in X.Consider a trivialisation Vi, ϕi of X which is of the form Vi = Ui×Bi,

where Ui = Vi ∩X0, ϕi|X0 and Bi, ϕi|B are complex atlases of X0 and B.

Assume that dimB = 1, so that OBε = C[ε](ε2). For all i, Θi : OXε |Vi 'OUi [ε]/(ε2) is an isomorphism. On Vi∩Vj , these identifications determine anautomorphism θi,j : : OXε |Vi∩Vj ' OUi∩Uj [ε]/(ε

2) which is compatible withπ, so that we have an exact sequence:

0→ OUij · ε→ OUij [ε]/(ε2)→ OUij → 0.

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By the definition of derivations (cf. Remark 1.36), the morphism of ringsθij corresponds to a derivation χij ∈ Γ(Uij , TUij ) of OUij by

φ ∈ OUij 7→ φ+ θij(φ)ε,

and by definition the χij satisfy the cocycle condition. A change of trivial-isation of X modifies χij by a coboundary, so that χij defines a (Cech)cohomology class in H1(X0, TX0).

Claim 5.9. The class of the cocycle χij is ρ( ∂∂ε).

Indeed, TBε,0 is the sheaf generated by the vector field ∂∂ε , and (TXε)|X is

the sheaf of free OX -modules generated by TX0 and ∂∂ε . On Ui, χi = θ∗i (

∂∂ε);

χi defines a splitting of the exact sequence:

0→ TX0 → (TXε)X0 → TBε,0 ⊗OX0 → 0.

On Ui, χi = ∂∂ε θi = θ∗i (

∂∂ε) and χij = χi−χj , and one checks–as in the

proof of Proposition 5.8– that ρ( ∂∂ε) = [χij ].

Remark 5.10. Conversely, an interesting question is to ask under whatconditions does a class θ ∈ H1(X,TX) define a deformation of the complexstructure of X. This is the object of the theory of obstructions, and as inthe case of integrability conditions, this leads to a non-linear condition.

In fact, if there exists a universal family of deformations of X over thegerm of an analytic set (B, 0), (B, 0) ' H1(X,TX) and the existence ofobstructed deformations corresponds to the presence of a singularity at 0 ∈B.

After this digression, let us go back to the study of a family of complexmanifolds π : X → B, where B is locally contractible and X0 = π−1(0) isthe central fibre. By Proposition 5.5–or even by Theorem 5.3–for all t ∈ Bin a neighbourhood of 0 ∈ B, Hk(Xt,C) ' Hk(X0,C).

In an appropriate trivialisation of X , B and π (with open sets of the formVi = Ui ×Bi as above),

V 7→ Hk(π−1V,G)

is determined by the Kunneth formula; where V ⊂ B is an open set andG = Z,Q,R or C.

Definition 5.11. Let G be an abelian group. A local system of stalk G isa sheaf of abelian groups/ vector spaces that is locally isomorphic to theconstant sheaf of stalk G.

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Definition 5.12. The Hodge bundles Hk = Rkπ∗G are the sheaves of freeOB-modules (resp. free AB-modules) for G = Z,Q,R or C assocciated to thepresheaves V 7→ Hk(π−1V,G); Rkπ∗G is the local system of stalkHk(X0, G).

Definition 5.13. The Gauss-Manin connection ∇ : Hk → Hk ⊗ ΩB is thenatural connection which coincides with the exterior differential d in a localtrivialisation of X , B and π as above.

We will need the following:

Lemma 5.14 (The Lie-Cartan formula). Let u ∈ Γ(TB,0) and v ∈ Γ((TX )X0)be a vector field such that π∗(v) = u. Let Ω ∈ AkX be such that Ωt = Ω|Xtis closed for all t ∈ B and denote ω : B → Hk(Xt,C) be the map t 7→ [Ωt].Then:

∇(ω)|0(u) = [i(v)(dΩ)X0 ],

where i(v) is the interior product with v as in (23), and [α] denotes the classof α.

Proof. Admitted.

5.2 The case of Kahler manifolds

When the central fibre of a complex family is Kahler, we obtain more preciseresults, and the Hodge Decomposition on the cohomology “deforms” withthe family.

Theorem 5.15. Let π : X → B be a family of complex manifolds. As-sume that the central fibre X = π−1(0) is Kahler and let F be a holomor-phic vector bundle over X . Then, the function b 7→ dimHq(Xb,F|Xb) isupper semi-continuous, i.e. in a neighbourhood of 0, dimHq(Xb,F|Xb) ≤dimHq(X,F|X).

Corollary 5.16. For b in a neighbourhood of 0 in B, the function b 7→hp,q(Xb) is upper semi-continuous and

Hk(Xb,C) =⊕p+q=k

Hp,q(Xb),

where Hp,q(Xb) = Hq,p(Xb) and Hp,q(Xb) = Hq(Xb,ΩpXb).

Remark 5.17. It is important to note that the statement of the corollaryis that a Hodge Decomposition holds on the cohomology of fibres near 0,but we do not know a priori whether these fibres are Kahler or not.

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The result will follow from the following proposition, which we admit.

Proposition 5.18 (Kodaira). Let π : X → B be a family of manifolds, andG→ X a vector bundle. Let ∆ be a relative differential operator acting onG, that is ∆b = ∆Xb : GXb → GXb is a differential operator. If each ∆b iselliptic of fixed order, then b 7→ dim ker ∆b is upper semi-continuous, andker ∆b ⊂ C∞(Gb) varies in a C∞ way, and forms a complex subbundle ofG.

Remark 5.19. Note that since in a suitable trivialisation Xπ−1(V ) ' U×V ,for U, V open sets of X and B respectively, b 7→ ∆b can be seen as a familyof differential operators acting on G|X → X.

Proof of Theorem 5.15. Endow F and X with hermitian metrics; these in-duce Hermitian metrics on Fb and Xb for all b ∈ B. Apply Proposi-tion 5.18 to the ∂-Laplacian, that acts on the sections A0,q

Xb (FXb). Here,

ker ∆b = H0,q

∂(FXb), by Corollary 3.11, H0,q(FXb) ' Hp,q(Xb) (note that we

only need Xb to be a compact complex manifold, which is the case becauseπ is proper and holomorphic) , and the result follows.

We may define a filtration on the cohomology of the fibres Xb withoutassuming that they are Kahler (however this filtration will not necessarilyhave the properties of a Hodge filtration) as follows. Recall that

F pAk(Xb) = α = Σiαi,k−i|αi,k−i = 0 for all i < p,

we define:

F pHk(Xb,C) = ker(d : F pAk → F p+1Ak)/ im(d : F p−1Ak → F pAk), (23)

so that F pHk(Xb,C) is the set of classes representable by closed forms thatare sums of forms of type (i, k − i) with i ≥ p.

When Xb is Kahler, we have seen that F pHk(Xb,C)/F p+1Hk(Xb,C) 'Hk(Xb,ΩXb). In general, F pHk(Xb,C)/F p+1Hk(Xb,C) is isomorphic to aquotient of Hq(Xb,Ωp

Xb).

Proof of Corollary 5.16. Apply Theorem 5.15 to the holomorphic vectorbundle Ωp

X/b = ΛpΩX/B, where ΩX/B is defined by the exact sequence:

0→ π∗ΩB → ΩX → ΩX/B → 0.

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By Theorem 5.3, bk = dimHk(X,C) = dimHk(Xb,C) for all b near 0.Since for all p, F pHk(Xb,C)/F p+1Hk(Xb,C) is isomorphic to a quotient ofHq(Xb,Ωp

Xb), we may write

bk = dimHk(Xb,C) = Σ dim(F pHk(Xb,C)/F p+1Hk(Xb,C)) ≤ Σhp,k−p(Xb).(24)

Theorem 5.15 shows that for all p, hp,k−p(Xb) ≤ hp,k−p(X), and henceΣhp,k−p(Xb) ≤ Σhp,k−p(X) = bk, so that all inequalities in (24) are equalitiesand this forces

F pHk(Xb,C)/F p+1Hk(Xb,C) ' Hq(Xb,ΩpXb)

for all p.

Proposition 5.20. For b in a neighbourhood of 0 ∈ B, there is a decompo-sition on the Betti cohomology of the fibre Xb:

Hk(Xb,C) =⊕

Hp,k−p(Xb,C),

where Hp,q(Xb) = Hq,p(Xb), and Hp,q(Xb) ' Hq(Xb,ΩpXb) for all p, q.

Sketch Proof. By Proposition 5.18, F pHk(Xb,C) ⊂ Hk(Xb,C) ' Hk(X,C)varies in a C∞ way, and the Hodge decomposition on Hk(X,C) yields that

F pHk(Xb,C) ∩ F k−p+1Hk(Xb,C) = Hp,k−p(Xb)

varies in a C∞ way, has constant dimension for b near 0, and hence isisomorphic to Hk−p(Xb,Ωp

Xb); the direct sum decomposition on Hk(Xb,C)follows.

The statement on complex conjugation and direct sum decomposition onthe cohomology of Xb follow from the Hodge Decomposition on Hk(X,C) bycontinuity (the dimensions of summands are constant in a neighbourhoodof 0 ∈ B).

Remark 5.21. In fact, more is true: Kodaira proves that if the central fibreX is Kahler, Xb is Kahler for all b in a neighbourhood of 0 ∈ B.

5.3 Period Domains and the Period map

We fix the setup for this section. Let π : X → S be a family of complexmanifolds, whose central fibre X is assumed to be Kahler, and where Bis a contractible neighbourhood of 0 ∈ B. By Theorem 5.3, the fibres of

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π are diffeomorphic, and hence Hk(Xb,C) ' Hk(X,C) for all b ∈ B, andwe have defined the local system Hk for all k (definition 5.12). Further,Theorem 5.15 shows that the function b 7→ dimF pHk(Xb,C) is constant onB (possibly shrinking B). Denote bp,k the value of dimF pHk(Xb,C) on B.

Definition 5.22. The period map of the family π : X → B is:

Pp,k : B → Gr(bp,k, Hk(X,C))

b 7→ (F pHk(Xb,C) ⊂ Hk(X,C))

By Proposition 5.18, we may in fact define complex subbundles F pHkover B, whose fibre over b ∈ B is F pHk(Xb,C). The period map is then themap:

Pp,k : b 7→ [(F pHk)b] ∈ Gr(bp,k, Hk(X,C)).

Recall (Example Sheet 1) that the Grassmannian G = Gr(k, n) of k-planes in Cn is a projective manifold of dimension k(n− k), and that thereis a tautological vector bundle S → G over G, whose fibre over a point[K] ∈ G is the subspace K ⊂ Cn.

We recall briefly a construction of charts for the Grassmannian G. Let[K] ∈ G and let V be a supplementary subspace of K, i. e. such that K⊕V =Cn. Denote GV the open set Z ⊂ Cn; dimZ = k and Z∩V = 0. DenoteπV and πK the projections of Cn on V and K; a complex chart on GV isdefined by identifying Z ∈ GV to the element hZ = πV (π−1

K )|Z : K → V ∈HomC(K,V ). This yields a canonical isomorphism:

TG,[K] ' THomC(K,V ),0 ' Hom(K,Cn/K)

Note that if s1, · · · , sk is a basis of K, and si : G → S are the holomorphicsections defined by si(K) = si, then if u ∈ TG,[K], hu ∈ Hom(K,Cn/K) isthe map

hu : K → Cn/K

such that hu(si) = u(si) mod K, where u(si) is the derivative of si alongu.

We now come to one of the main results of this course.

Theorem 5.23. The period map Pp,k : B → Gr(bp,k, Hk(X,C)) is holomor-phic for all p, k ≥ 0.

Proof. It is clear from Definition 5.22 that the period map Pp,k is C∞ (thisfollows from Proposition 5.18). It is enough to prove that dPp,k is C-linear,or equivalently that dPp,k(u) = 0 for all u ∈ TB,b,C of type (0, 1).

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Step 1. From what we have just seen on tangent spaces of Grassmanni-ans, the differential of Pp,k is a linear map

dPp,k : TB,b → Hom(F pHk(Xb,C), Hk(Xb,C)/F pHk(Xb,C)).

Let s1, · · · , sk be a basis of F pHk(Xb,C) and denote si the associated holo-morphic sections of the tautological line bundle S → Gr. We may see sias C∞-sections of Hk over B with si(b) = si, and si(b

′) ∈ F pHk(Xb′ ,C)for all b′ in a neighbourhood of b ∈ B (in other words: si are C∞-sectionsof F pHk). Then, dPp,k(u)(si) = u(si) mod F pHk(Xb,C), or equivalently,dPp,k(u)(si) = ∇u(si), where ∇ is the Gauss-Manin connection.

Step 2. We want to compute the directional derivative ∇u(si). Forthis, denote T : X ' X ×B a diffeomorphism as in Proposition 5.5, so thatT : x×B → X is holomorphic for all x ∈ X. By Proposition 5.18, possiblyafter shrinking B, the subspace of Hk(Xb′ ,C) ' Hk(X,C) consisting ofcohomology classes represented by elements of F pAkXb′ is of constant rank,

there is a differential form Ω ∈ F pAkX/B– or after lifting Ω ∈ F pAkX such

that Ω|Xb′ is closed for all b′ ∈ B and [Ω|Xb′ ] = si(b′). We view Ω as a section

b′ 7→ si(b′) of Hk.

As we have seen, T defines a C∞ splitting of

0→ (TX/B)|Xb → (TX )|Xb → (π∗TB)|Xb → 0

that defines a direct sum decomposition (TX )|Xb = TXb ⊕ M with M 'π∗(TB,b). This decomposition remains valid on the real tangent bundle be-cause the map π is holomorphic.

Pick v a section of M such that π∗v = u. The Lie-Cartan formula(Lemma 5.14) implies

∇(si)|b(u) = [i(v)(dΩ)Xb,

so that

dPp,k(u)(si) = [i(v)(dΩ)|Xb ] mod F pHk(Xb,C).

The form dΩ ∈ F pAk+1X , so that the class of i(v)(dΩ)|Xb is in F pHk(Xb,C). If

we assume that u ∈ T 0,1B,b,C, then the restriction of v to Xb is of type (0, 1), and

dPp,k(u)(si) = 0 for i = 1, · · · , k. This shows that Pp,k is holomorphic.

Definition 5.24. For all p, k ≥ 0, there is a holomorphic vector bundle

F pHk ⊂ Hk → B

89

with fibre F pHk(Xb,C), defined as (Pp,k)∗S, where S → Gr(bp,k, F pHk(X,C))is the tautological vector bundle.

For all k ≥ 0, there is a filtration

· · · ⊂ F pHk ⊂ F p−1Hk ⊂ · · · ⊂ F 0Hk = Hk

of the Hodge bundle by holomorphic sub-bundles.

DefineHp,k−p = F pHk/F p+1Hk as the successive quotients of these holo-morphic vector bundles. When X is Kahler, Corollary 5.16 shows that thefibre over b ∈ B is Hp,k−pb = Hk−p(Xb,Ωp

Xb).

Remark 5.25. We have already mentioned the vector bundles F pHk; it isa consequence of Theorem 5.23 that they are in fact holomorphic. Comparethis definition with E.S. 2, Ex.6.

Theorem 5.26 (Griffiths’ transversality). The differential of the period map

dPp,k : TB,b → Hom(F pHk(Xb,C), Hk(Xb,C)/F pHk(Xb,C))

takes values in Hom(F pHk(Xb,C), F p−1Hk(Xb,C)/F pHk(Xb,C)).

Proof. We use the notation of the proof of Theorem 5.23. As in Step 2.of the proof of Theorem 5.23, for all u ∈ TB,b, i(v)(dΩ)|Xb is a differential

form in F p−1AkX , and dPp,k(u)(si) ∈ F p−1Hk(Xb,C) for all i. The resultfollows.

Corollary 5.27. The holomorphic subbundles F pHk satisfy the transver-sality property:

∇F pHk ⊂ F p−1Hk.

Proof. This is a direct consequence of Theorem 5.26. Indeed, a holomorphicsection s of F pHk is in particular a differentiable section of Hk such thats(b) ∈ F pHk(Xb,C) for all b ∈ B, and for all u ∈ TB,b,

dPp,k(u)(s(b)) = ∇u(s) mod F pHk(Xb,C).

Definition 5.28. The flag space Fb•,W associated to a (k + 1)-tuple ofintegers b• = (bi)0≤i≤k with bi ≤ bi−1 and to a C-vector space W ' Cn is:

Fb•,W = (W0, · · · ,Wk) ∈ Π Gr(bi,W )|W i ⊂W i−1.

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Exercise 5.29. Check that Fb•,W is a complex manifold and that its tangentspace at [W ] = (W 0, · · · ,W k) is:

TFb•,W ,[W ] = (h0, · · · , hk) ∈ ⊕Hom(W i,W/W i)|(hi)|W i+1 = hi+1 mod W i.

Definition 5.30. For all k ≥ 0, the map P = (P1, · · · ,Pk)

P : b ∈ B 7→ [(F 0Hk(Xb,C), · · · , F kHk(Xb,C))] ∈ Fbp,k,Hk(X,C)

is holomorphic. The open set D = imP in Fbp,k,Hk(X,C) is the non-polarisedperiod domain.

Exercise 5.31. Determine the flag spaces associated to weight 1 and 2 Hodgestructures on a Kahler manifold.

Assume further that there is a class [ω] ∈ H2(X ,Z) such that ω|Xb isa Kahler form and its class is locally constant for b ∈ B (recall that B isassumed to be a contractible neighbourhood of 0). Cup product with theclass [ω] defines a global Lefschetz operator, that is a morphism of localsystems (this is a morphism of Hodge Structures of type (1, 1)):

L : Rkπ∗C→ Rk+2π∗C or, equivalently: L : Hk → Hk+2.

We generalize the Lefschetz decomposition (see Lemma 3.39) to a relativesetting as follows.

Let n = dimXb and define:

(Riπ∗C)prim = kerLn−i+1,

then there is a Lefschetz Decomposition:

Rkπ∗C =⊕

2k−2n≤2r≤kLr(Rk−2rπ∗C)prim.

The restriction to each fibre of this decomposition is compatible with theHodge Decomposition.

The intersection form Q associated to the polarisation ωXb is globalisedin the obvious way (i. e. Q(α, β) =< Ln−kα, β > for forms of the appropriatedegree on X ).

There is a polarised period map Ppol associated to the polarised Hodgestructure Hk(Xb,C)prim for all k ≥ 0 that keeps track of the Hodge decom-position and of the intersection form on the cohomology of the fibres. Theimage of Ppol is the polarised period domain Dpol.

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Example 5.32 (ES 6, ex.3). Let X be a compact Kahler manifold and H =H1(X,C) = H1,0(X) ⊕ H0,1(X). If X is polarised, there is an alternatingintersection form

Q : H ×H → Z,

such that for all x ∈ F 1H, Q(x, x) = 0 and Q(x, x) > 0. These two condi-tions imply that the polarised period domain Dpol is an open set in a quadrichypersurface.

We now turn to the infinitesimal study of the period map; i.e. we willstudy variations of Hodge structures. We now draw some consequences ofTheorem 5.26 and Corollary 5.27. Consider the following diagram, which

defines the operator ∇p,k−p:

F p+1Hk ∇ //

F pHk ⊗ ΩB

F pHk ∇ //

F p−1Hk ⊗ ΩB

Hp,k−p ∇p,k−p//

Hp−1,k−p+1 ⊗ ΩB

0 0

Definition 5.33. The operator ∇p,k−pb : Hp,k−pb → Hp−1,k−p+1b ⊗ΩB,b is the

Infinitesimal Variation of Hodge Structure at b ∈ B.

Lemma 5.34. The differential dPp,kb : TB,b → TGr,F pHk(Xb) is the map con-structed by adjunction from

∇p,k−pb : Hp,qb → Hp−1,q+1b ⊗ ΩB,b.

Proof. This is a manipulation of the definitions and a consequence of Corol-lary 5.27: check that

dPp,k : TB,b → Hom(F pHk(Xb)/F p+1Hk(Xb), F p−1Hk(Xb)/F pHk(Xb))TB,b → Hom(Hq(Xb,Ωp

Xb), Hq+1(Xb,Ωp−1

Xb ))

is the map obtained from ∇p,k−pb by adjunction.

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Theorem 5.35 (Griffiths). Let u be a tangent vector in TB,b, the map

dPp,k(u) : Hq(Xb,ΩpXb)→ Hq+1(Xb,Ωp−1

Xb ))

is the map obtained as the composition of the cup product with the Kodaira-Spencer class with the map induced on cohomology by the interior product

TXb ⊗ ΩpXb → Ωp−1

Xb .

Remark 5.36. Let X be a complex manifold and E,F holomorphic vectorbundles on X. Denote E ,F the associated sheaves of free OX -modules. Thecup-product

∪ : Hr(X, E)⊗Hs(X,F)→ Hr+s(X, E ⊗ F)

is the map on cohomology induced by the exterior product of forms on Dol-beault cohomology representatives. Recall from Example 1.83 the Dolbeaultresolution of the sheaves E , F , the cup product is the map on cohomologyinduced by the natural map:

A0,r(E)⊗A0,s(F )→ A0,r+s(E ⊗ F ).

Proof. We have seen in the proof of Proposition 5.8 that if u ∈ TB,b, the classρ(u) is represented by ∂v|Xb , where v is a C∞ vector field on TX such thatπ∗v = u and v|Xb is of type (1, 0). Here, we assume that u ∈ TB is a holo-morphic tangent vector, i.e. when viewed as an element of the complexifiedtangent bundle, u is of type (1, 0).

If σ ∈ Hk−p(Xb,ΩXb), by Proposition 5.18, there is a form Ω ∈ F pAXsuch that the restriction Ω|Xb′ = Ωb′ is closed for all b′ ∈ B and whoseDolbeault cohomology class [Ωb] = σ. Then, as in the proof of Theorem 5.23,Lemma 5.14 implies that:

dPp,k(u)(σ) = ∇p,k−pu (σ) = [i(v)(dΩ)p−1,k−p+1|Xb ].

The closed form i(v)(dΩ)|Xb lies in F p−1AkXb , and the Dolbeault cohomology

class of the component of type (p− 1, k − p+ 1) lies in Hk−p+1(Xb,Ωp−1Xb ).

Since v|Xb is of type (1, 0), we have the following equalities for the com-ponent of type (p− 1, k − p+ 1) of i(v)(dΩ)|Xb :

i(v)(dΩ)p−1,k−p+1|Xb = i(v)(∂Ω)p−1,k−p+1

|Xb = i(v)(∂Ωp,k−p)|Xb .

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Since i(v)(∂Ωp,k−p)− i(∂v)(Ωp,k−p) = −∂(i(v)(Ωp,k−p)), and i(v)(Ωp,k−p) isa ∂-closed form, restricting to Xb gives:

dPp,k(u)(σ) = [i(∂v)(Ωp,q|Xb)] = [ρ(u) ∪ σ].

Remark 5.37. Recall that Hp,k−pb = Hk−p(Xb,ΩpXb). Theorem 5.35 shows

that:

∇p,k−pb : Hp,k−pb → Hp−1,k−p+1b ⊗ ΩB,b = Hom(TB,b,Hp−1,k−p+1

b )

can be identified with the composition of

Hk−p(Xb,ΩpXb)→ Hom(H1(Xb, TXb), H

k−p+1(Xb,Ωp−1Xb )

given by cup product with

ρ∗ : Hom(H1(Xb, TXb), Hk−p+1(Xb,Ωp−1

Xb )→ Hom(TB,b, Hk−p+1(Xb,Ωp−1

Xb ).

The Torelli theorem for curves We admit some results on the defor-mation theory of compact complex manifolds.

Definition 5.38. Let X be a projective manifold. A deformation of Xover a germ (S, 0) of complex manifold is a family of complex manifoldsπ : X → S, with π−1(0) = X (here π is proper, flat and submersive).

A family π : X → S is a universal family of deformations of X if forall deformation π′ : X ′ → S′, there is a unique map S′ → S such that thediagram

X ′ //

π′

Xπ

S′ // S

where X ′ → X is the map induced by S → S′. The base S is the Kuranishispace Def(X) of X; X is the Kuranishi family of X.

Theorem 5.39 (Kuranishi theorem). Let X be a compact complex manifold.If H0(X,TX) = (0), there is a universal deformation space Def(X) = Swith bijective Kodaira-Spencer map.The base S is the fibre over the ori-gin of a holomorphic map between neighbourhoods of the origin in 0 ∈H1(X,TX)→ 0 ∈ H2(X,TX) with vanishing differential at 0. The base Sis smooth precisely when this map is zero.

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Let X be a compact Riemann surface (a compact complex manifold ofdimension 1) and denote g = dimH0(X,Ω1

X). When g ≥ 2, H0(X,TX) =H2(X,TX) = (0), and Kuranishi’s theorem shows that there is a universaldeformation family π : X → S over a smooth germ (S, 0). Since the Kodaira-Spencer map is bijective, TS ' H1(X,TX).

Remark 5.40. It is known that there also exists a universal family of de-formations X → S when the genus is 0 or 1 (Compare with ES 2, ex.3 forthe case of g = 1)).

Question 5.41 (Infinitesimal Torelli problem). Assume that X → S is afamily of complex manifolds over a contractible base, with injective Kodaira-Spencer map ρ (we will focus on the universal family of deformations of Xin what follows).

The infinitesimal Torelli problem is concerned with knowing when theperiod map P : S → D is an immersion. This amounts to asking whetherinfinitesimal changes of the complex structure at 0 are faithfully reflected inthe changes of the Hodge Decomposition near the central fibre.

By Theorem 5.35, finding a criterion for P to be an immersion is equiv-alent to finding a criterion for the cup-product

H1(X,TX)→ ⊕Hom(Hq(X,ΩpX), Hq+1(X,Ωp−1

X ))

to be injective on TS,0 (recall that the Kodaira-Spencer map TS,0 → H1(X,TX)is assumed to be injective).

Assume that X is a compact complex curve, and let π : X → S beits universal family of deformations of X. The study of the period mapP = (P0,P1,P2) is equivalent to the study of P1 : S → Gr(g,H1(X,C)).We want to determine criteria for P1 to be an immersion near s ∈ S, thatis for

dP1 : TS,s → Hom(H1,0(Xs), H0,1(Xs))to be injective. By Theorem 5.35, this is equivalent to determining whenthe map

H1(Xs, TXs)→ Hom(H0(Xs,ΩXs), H1(Xs,OXs))given by cup product and the contraction

H0(Xs,ΩXs)⊗H1(Xs, TXs)→ H1(Xs,OXs) (25)

is injective.Recall that TXs is dual to KXs = ΩXs . Further, by Serre Duality,

H1(Xs,OXs) ' H0(Xs,KXs)∗ and H1(Xs, TXs) ' H0(Xs,KXs⊗KXs)∗ (26)

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Lemma 5.42. The map

H0(Xs,KXs)⊗H0(Xs,KXs)→ H0(Xs,KXs ⊗KXs)

obtained by dualising (25) and applying Serre Duality coincides with themultiplication of sections µ.

Proof. Let η = α⊗β be a global section ofK⊗2Xs ( i.e. η ∈ H0(Xs,KXs⊗KXs)).

Consider a class [u] ∈ H1(Xs), TXs , where the representative u is a sectionu ∈ A0,1(TXs). We have:

< µ(η), u >= [uαβ] ∈ H1(Xs,KXs) ' C,

where the class [uαβ] is obtained by contracting u ∈ A0,1(TXs) with thesection α⊗ β of KXs ⊗KXs . We also have [uαβ] = [(uα)β], where the classof uα is in H1(Xs,OXs) and the class of β is in H0(Xs,KXs). This provesthe Lemma.

Definition 5.43. A complete curve X is hyperelliptic if there exists a 2 : 1rational map X → P1; X is trigonal if there is a 3 : 1 rational map X → P1.

We admit the following results, which are consequences of the classicalstudy of algebraic curves.

Theorem 5.44. 1. A generic complete curve X of genus g ≥ 3 (resp. g ≥5) is not hyperelliptic (resp. not trigonal). A generic curve of genusg = 6 is not a planar quintic (i.e. a curve of the form X = f5 = 0 ⊂P2 where f5 is a homogeneous polynomial of degree 5).

2. (Noether) If X is a non-hyperelliptic curve, the multiplication map µis surjective.

3. (Petri) If X is non-hyperelliptic, non-trigonal, and not a planar quin-tic, X is determined by kerµ.

Corollary 5.45 (Infinitesimal Torelli theorem for curves). Let X be a non-hyperelliptic curve. The local period map

P1 : S → Gr(g,H1(X,C))

is an embedding at the point 0 ∈ S corresponding to X.

Proof. By Lemma 5.42, when µ is surjective, the period map P1 is an em-bedding. The result follows from Noether’s theorem.

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As a result, when X is a generic complete curve of genus g ≥ 3, P1 is anembedding, so that variations of the complex structure on X are faithfullyrepresented by variations of the Hodge structure.

One can show that when X is not hyperelliptic, the canonical morphism(associated to the linear system KX)

Φ|KX | : X → Pg−1 = P(H0(X,ΩX))

is an embedding. Petri’s result claims that if X is neither trigonal nor aplanar quintic, symmetric elements of kerµ are precisely the homogeneouspolynomials of degree 2 on Pg−1 that vanish on the image of X. In otherwords, the image of X in Pg−1 is the algebraic subscheme of Pg−1 definedas the intersection of quadric hypersurfaces that contain it (this is a highlynon-trivial statement).

Theorem 5.46 (Generic Torelli Theorem for curves). If X is a genericcompact complex curve of genus g ≥ 5, X is determined by its infinitesimalvariation of Hodge structure and by the isomorphism

H1,0(X) ' (H1(X,C)/H1,0(X))∗.

In particular, let X and X ′ be two compact complex curves. Assume thati : H1(X,C) ' H1(X ′,C) is an isomorphism compatible with the intersectionforms, that j : (S, 0) ' (S′, 0) is a germ isomorphism between the bases of thelocal universal deformations of X and X ′, and that j yields an identificationof the variations of Hodge structures in the neighbourhood of X and X ′:

P1 : S //

j

Gr(g,H1(X,C))

i

P1 : S′ // Gr(g,H1(X ′,C))

,

then the curves X and X ′ are isomorphic.

Proof. We prove the first statement: we have seen that the infinitesimalvariation of Hodge structure dP1 at 0 ∈ S defines a map that is symmetric(relative to the Serre Duality H1,0(X) ' (H1(X,C)/H1,0(X))∗):

TS,0 → Hom(H1,0(X), H1(X,C)/H1,0(X)).

Given an isomorphism H1,0(X) ' (H1(X,C)/H1,0(X))∗, we may dualisethis map to a (symmetric) map:

µ : H1,0(X)⊗H1,0(X)→ (TS,0)∗.

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By Theorem 5.44, when X is non-hyperelliptic, non-trigonal and not a pla-nar quintic, X is identified with the subscheme of P(H1,0(X)∗) defined bysymmetric elements of kerµ.

The second statement follows at once. By differentiation, the diagramabove gives an identification of the infinitesimal variations of Hodge struc-tures at s and j(s) for all s ∈ S, that is compatible with the Serre dualityisomorphisms (because i preserves the intersection form), so that the verticalarrows in

µ∗s : TS,s //

j∗

Hom(H1,0(Xs), (H1(Xs,C)/H1,0(Xs))∗

i∗

µ∗j(s) : TS′,j(s) // Hom(H1,0(X ′j(s)), (H1(X ′j(s),C)/H1,0(X ′j(s)))

∗

are isomorphisms. For generic s ∈ S, Xs ' X ′j(s) because Xs and X ′j(s) aredetermined by kerµs and kerµj(s) respectively. It then can be proved thatXs ' X ′j(s) for all s ∈ S.

Remark 5.47. The generic Torelli theorem for curves of genus g ≥ 5 as-serts that the global period map that associates to X the polarised HodgeStructure on H1(X,Z) is of degree 1 on its image. This is a finer statement:it is known that the global polarised period map is injective, so that if twocomplex curves have isomorphic polarised Hodge structures, they are iso-morphic. Note that such strong statements do not in general hold in higherdimensions.

References

[Dem96] Jean-Pierre Demailly. Theorie de Hodge L2 et theoremesd’annulation. In Introduction a la theorie de Hodge, volume 3 ofPanor. Syntheses, pages 3–111. Soc. Math. France, Paris, 1996.

[GH78] Phillip Griffiths and Joseph Harris. Principles of algebraic geom-etry. Wiley-Interscience [John Wiley & Sons], New York, 1978.Pure and Applied Mathematics.

[Huy05] Daniel Huybrechts. Complex Geometry. Universitext. Springer-Verlag, Berlin, 2005.

[Voi02] Claire Voisin. Hodge Theory and Complex Algebraic Geometry.I-II, volume 76,77 of Cambridge Studies in advanced mathematics.Cambridge University Press, Cambridge, 2002.

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