Hodge Theory

Hodge Theory

Complex Manifolds

by William M. Faucette

Adapted from lectures by

Mark Andrea A. Cataldo

Structure of Lecture

Conjugations

Tangent bundles on a complex manifold

Cotangent bundles on a complex manifold

Standard orientation of a complex manifold

Almost complex structure

Complex-valued forms

Dolbeault cohomology

Conjugations

Conjugations

Let us recall the following distinct notions of conjugation.

First, there is of course the usual conjugation in C:

Conjugations

Let V be a real vector space and

be its complexification. There is a natural R-linear isomorphism given by

Tangent Bundles on a Complex Manifold

Tangent Bundles on a Complex Manifold

Let X be a complex manifold of dimension n, x2X and

be a holomorphic chart for X around x. Let zk=xk+iyk for k=1, . . . , N.

Tangent Bundles on a Complex Manifold

TXR) is the real tangent bundle on X. The fiber TX,xR) has real rank 2n and it is the real span

Tangent Bundles on a Complex Manifold

TXC):= TXR)RC is the complex tangent bundle on X. The fiber TX,xC) has complex rank 2n and it is the complex span

Tangent Bundles on a Complex Manifold

Often times it is more convenient to use a basis for the complex tangent space which better reflects the complex structure. Define

Tangent Bundles on a Complex Manifold

With this notation, we have

Tangent Bundles on a Complex Manifold

Clearly we have

Tangent Bundles on a Complex Manifold

In general, a smooth change of coordinates does not leave invariant the two subspaces

Tangent Bundles on a Complex Manifold

However, a holomorphic change of coordinates does leave invariant the two subspaces

Tangent Bundles on a Complex Manifold

TX is the holomorphic tangent bundle on X. The fiber TX,x has complex rank n and it is the complex span

TX is a holomorphic vector bundle.

Tangent Bundles on a Complex Manifold

TX is the anti-holomorphic tangent bundle on X. The fiber TX,x has complex rank n and it is the complex span

TX is an anti-holomorphic vector bundle.

Tangent Bundles on a Complex Manifold

We have a canonical injection and a canonical internal direct sum decomposition into complex sub-bundles:

Tangent Bundles on a Complex Manifold

Composing the injection with the projections we get canonical real isomorphisms

Tangent Bundles on a Complex Manifold

The conjugation map

is a real linear isomorphism which is not complex linear.

Tangent Bundles on a Complex Manifold

The conjugation map induces real linear isomorphism

and a complex linear isomorphism

Cotangent Bundles on Complex Manifolds

Cotangent Bundles on Complex Manifolds

Let {dx1, . . . , dxn, dy1, . . . , dyn} be the dual basis to {x1, . . . , xn, y1, . . . , yn}. Then

Cotangent Bundles on Complex Manifolds

We have the following vector bundles on X:

TX*(R), the real cotangent bundle, with fiber

Cotangent Bundles on Complex Manifolds

TX*(C), the complex cotangent bundle, with fiber

Cotangent Bundles on Complex Manifolds

TX*(C), the holomorphic cotangent bundle, with fiber

Cotangent Bundles on Complex Manifolds

TX*(C), the anti-holomorphic cotangent bundle, with fiber

Cotangent Bundles on Complex Manifolds

We have a canonical injection and a canonical internal direct sum decomposition into complex sub-bundles:

Cotangent Bundles on Complex Manifolds

Composing the injection with the projections we get canonical real isomorphisms

Cotangent Bundles on Complex Manifolds

The conjugation map

is a real linear isomorphism which is not complex linear.

Cotangent Bundles on Complex Manifolds

The conjugation map induces real linear isomorphism

and a complex linear isomorphism

Cotangent Bundles on Complex Manifolds

Let f(x1,y1,…, xn, yn)= u(x1,y1,…, xn, yn)+ i v(x1,y1,…, xn, yn) be a smooth complex-valued function in a neighborhood of x. Then

The Standard Orientation of a Complex Manifold

Standard Orientation

Proposition: A complex manifold X admits a canonical orientation.

If one looks at the determinant of the transition matrix of the tangent bundle of X, the Cauchy-Riemann equations immediately imply that this determinant must be positive.

Standard Orientation

If (U,{z1,…,zn}) with zj=xj+i yj, the real 2n-form

is nowhere vanishing in U.

Standard Orientation

Since the holomorphic change of coordinates is orientation preserving, these non-vanishing differential forms patch together using a partition of unity argument to give a global non-vanish- ing differential form.

This differential form is the standard orientation of X.

The Almost Complex Structure

Almost Complex Structure

The holomorphic tangent bundle TX of a complex manifold X admits the complex linear automorphism given by multiplication by i.

Almost Complex Structure

By the isomorphism

We get an automorphism J of the real tangent bundle TX(R) such that J2=-Id. The same is true for TX* using the dual map J*.

Almost Complex Structure

An almost complex structure on a real vector space VR of finite even dimension 2n is a R-linear automorphism

Almost Complex Structure

An almost complex structure is equivalent to endowing VR with a structure of a complex vector space of dimension n.

Almost Complex Structure

Let (VR, JR) be an almost complex structure. Let VC:= VRRC and JC:= JRIdC: VC VC be the complexification of JR.

The automorphism JC of VC has eigenvalues i and -i.

Almost Complex StructureThere are a natural inclusion and a natural

direct sum decomposition

where the subspace VRVC is the fixed locus of the

conjugation map associated with the complexification.

Almost Complex Structure V and V are the JCeigenspaces

corresponding to the eigenvalues i and -i, respectively,

since JC is real, that is, it fixes VRVC, JC

commutes with the natural conjugation map and V and V are exchanged by this conjugation map,

Almost Complex Structure there are natural R-linear isomorphisms

coming from the inclusion and the projections to the direct summands

and complex linear isomorphisms

Almost Complex Structure The complex vector space defined by

the complex structure is C-linearly isomorphic to V.

Almost Complex Structure

The same considerations are true for the almost complex structure (VR*, JR*). We have

Complex-Valued Forms

Complex-Valued Forms

Let M be a smooth manifold. Define the complex valued smooth p-forms as

Complex-Valued Forms

The notion of exterior differentiation extends to complex-valued differential forms:

Complex-Valued Forms

Let X be a complex manifold of dimension n, x2X, (p,q) be a pair of non-negative integers and define the complex vector spaces

Complex-Valued Forms

There is a canonical internal direct sum decomposition of complex vector spaces

Complex-Valued Forms

Definition: The space of (p,q)-forms on X

is the complex vector space of smooth sections of the smooth complex vector bundle p,q(TX*).

Complex-Valued Forms

There is a canonical direct sum decomposition

and

Complex-Valued Forms

Let l=p+q and consider the natural projections

Define operators

Complex-Valued Forms

Note that

Also,

Dolbeault Cohomology

Dolbeault Cohomology

Definition: Fix p and q. The Dolbeault complex is the complex of vector spaces

Dolbeault Cohomology

The Dolbeault cohomology groups are the cohomology groups of the complex

Dolbeault Cohomology

That is,

Dolbeault CohomologyTheorem: (Grothendieck-Dolbeault Lemma)

Let q>0. Let X be a complex manifold and u2Ap,q(X) be such that du=0. Then, for every point x2X, there is an open neighborhood U of x in X and a form v2Ap,q-1(U) such that

Dolbeault Cohomology

The Grothendieck-Dolbeault Lemma guarantees that Dolbeault cohomology is locally trivial.

Dolbeault CohomologyFor those familiar with sheaves and sheaf

cohomology, the Dolbeault Lemma tells us that the fine sheaves Ap,q

X of germs of C (p,q)-forms give a fine resolution of the sheaf p

X of germs of holomorphic p-forms on X. Hence, by the abstract deRham theorem