nLab
Dolbeault cohomology

under construction

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Complex geometry

Contents

Idea

Dolbeault cohomology of a complex manifold XX is the chain cohomology of the Dolbeault complex of XX (see there for more).

Properties

Dolbeault theorem

By the Dolbeault theorem (see there) Dolbeault cohomology is equivalently the abelian sheaf cohomology H q(X;Ω X p)H^q(X;\Omega_X^p), of the abelian sheaf Ω X p\Omega_X^p is Dolbeault complex of holomorphic p-forms.

Hodge isomorphism

For XX a Hermitian manifold write p,q(X)\mathcal{H}^{p,q}(X) for the space of (p,q)(p,q)-harmonic differential forms and write H p,qH^{p,q} for its Dolbeault cohomology in the bidegree.

Proposition

There is a canonical homomorphism

p,q(X)H p,q(X). \mathcal{H}^{p,q}(X) \longrightarrow H^{p,q}(X) \,.

If XX is compact, then this is an isomorphism, the Hodge isomorphism

Also

k(X,)H dR k(X,) \mathcal{H}^k(X,\mathbb{C}) \longrightarrow H^k_{dR}(X,\mathbb{C})

is an isomorphism

(e.g. Maddock, prop. 4.2.7)

Serre duality

Serre duality: On a Hermitian manifold XX of complex dimension dim (X)=ndim_{\mathbb{C}}(X) = n the Hodge star operator induces isomorphisms

p,q(X) np,nq(X). \mathcal{H}^{p,q}(X)\simeq \mathcal{H}^{n-p, n-q}(X) \,.

(e.g. Maddock, prop. 4.2.8)

Hodge decomposition

If XX is a compact Kähler manifold then

k(X)p+q=k p,q(X) \mathcal{H}^k(X) \simeq \underset{p+q = k}{\oplus} \mathcal{H}^{p,q}(X)

(e.g. Maddock, prop. 4.2.9)

This is called the Hodge decomposition.

Hodge symmetry

Over complex manifolds XX, Hodge symmetry is the property that the Dolbeault cohomology groups H p,q(X)H^{p,q}(X) are taken into each other under complex conjugation followed by switching the bidegree:

H p,q(X)H q,p(X)¯. H^{p,q}(X) \simeq \overline{H^{q,p}(X)} \,.

In particular this means that the dimension of the cohomology groups in degree (p,q)(p,q) coincides with that in bidegree (q,p)(q,p).

Dolbeault resolution

Given any holomorphic vector bundle EE, one can form the Dolbeault resolution EΩ 0,qE \otimes \Omega^{0,q}, where Ω 0,q\Omega^{0,q} is the sheaf of C C^\infty (0,q)(0,q)-forms. This is an acyclic resolution of EE and hence computes its sheaf cohomology.

(…)

Double complex and Frölicher spectral sequence

The Dolbeault complexes naturall fit into a double complex

Ω p,0 ¯ Ω p1,1 ¯ Ω p+1,0 ¯ Ω p,1 ¯ . \array{ \Omega^{p,0} &\stackrel{\bar \partial}{\to}& \Omega^{p-1,1} &\stackrel{\bar \partial}{\to}& \cdots \\ \downarrow^{\mathrlap{\partial }} && \downarrow^{\mathrlap{\partial }} \\ \Omega^{p+1,0} &\stackrel{\bar \partial}{\to}& \Omega^{p,1} &\stackrel{\bar \partial}{\to}& \cdots \\ \downarrow^{\mathrlap{\partial }} && \downarrow^{\mathrlap{\partial }} \\ \vdots && \vdots } \,.

The corresponding spectral sequence of a double complex is called the Frölicher spectral sequence. On a Kähler manifold it exhibits the Hodge filtration.

References

  • Zachary Maddock, Dolbeault cohomology (pdf)

Revised on June 16, 2014 03:31:08 by Urs Schreiber (88.128.80.80)