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Dolbeault complex

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differential geometry

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Idea

The Dolbeaul complex is the analog of the de Rham complex in complex geometry.

Definition

Dolbeault complex

On a complex manifold XX the de Rham complex Ω (X)\Omega^\bullet(X) refines to a bigraded complex Ω ,(X)\Omega^{\bullet, \bullet}(X), where a differential form of bidegree (p,q)(p,q) has holomorphic degree pp and antiholomorphic degree qq, hence is given on a local coordinate chart by an expression of the form

ω=f IJdz i 1dz i pdz¯ j 1dz¯ j q. \omega = \sum f_{I J} d z_{i_1} \wedge \cdots d z_{i_p} \wedge d \bar z_{j_1} \wedge \cdots \wedge d \bar z_{j_q} \,.

Moreover, the de Rham differential d\mathbf{d} decomposes as

d=+¯, \mathbf{d} = \partial + \bar \partial \,,

where :Ω ,Ω +1,\partial \colon \Omega^{\bullet, \bullet}\to \Omega^{\bullet + 1, \bullet} and ¯:Ω ,Ω ,+1\bar \partial \colon \Omega^{\bullet, \bullet}\to \Omega^{\bullet, \bullet + 1}.

The Dolbeault complex of XX is the chain complex (Ω ,(X),¯)(\Omega^{\bullet, \bullet}(X), \bar \partial). The Dolbeault cohomology of XX is the cochain cohomology of this complex.

Holomorphic differential forms

Here Ω p,0(X)\Omega^{p,0}(X) defines a holomorphic vector bundle and a holomorphic section is a differential form with local expression as above, such that the coefficient functions f IJf_{I J} are holomorphic functions. This is called a holomorphic differential form.

For p<dim (X)p \lt dim_{\mathbb{C}}(X) equivalently this is a differential form in the kernel of the antiholomorphic Dolbeaul operator ¯\bar \partial.

Properties

Dolbeault theorem

The complex analog of the de Rham theorem is the Dolbeault theorem:

for XX a complex manifold then ints Dolbeault cohomology in bi-degree (p,q)(p,q) is naturally isomorphic to the abelian sheaf cohomology in degree qq of the abelian sheaf Ω pΩ p,0\Omega^p \coloneqq \Omega^{p,0} of holomorphic p-forms

H p,q(X)H q(X,Ω p). H^{p,q}(X)\simeq H^q(X,\Omega^p) \,.

(…)

Let Disk complDisk_{compl} be the category of complex polydiscs in n\mathbb{C}^n and holomorphic functions between them.

For pp \in \mathbb{N} write Ω p:Disk complex opSet\Omega^p \colon Disk_{complex}^{op} \to Set for the sheaf of holomorphic differential p-forms.

Proposition

For XX a complex manifold, let {U iX}\{U_i \to X\} be a holomorphic good open cover. Then the Cech cohomology of this cover with coefficients in Ω p\Omega^p in degree qq is the Dolbeault cohomology in bidegree (p,q)(p,q)

H p,q(X)π 0sPSh(Disk comp)(C({U i},Ω p[q])). H^{p,q}(X) \simeq \pi_0 sPSh(Disk_{comp})(C(\{U_i\}, \Omega^p[q])) \,.

For instance (Maddock, theorem 1.0.1).

On Stein manifolds

Proposition

(Cartan theorem B)

For XX a Stein manifold,

H k(Ω p,(X),¯)={0 k0 Ω hol p(X) k=0. H^k(\Omega^{p,\bullet}(X), \bar \partial) = \left\{ \array{ 0 & k \neq 0 \\ \Omega^p_{hol}(X) & k = 0 } \right. \,.

For instance (Gunning-Rossi).

Proposition

For XX a Stein manifold of complex dimension nn, the compactly supported Dolbeault cohomology is

H k(Ω c p,(X),¯)={0, kn (Ω hol np(X)) *, H^k(\Omega_c^{p, \bullet}(X), \bar \partial) = \left\{ \array{ 0 , & k \neq n \\ (\Omega_{hol}^{n-p}(X))^\ast } \right. \,,

where on the right () *(-)^\ast denotes the continuous linear dual.

First noticed in (Serre).

Todd genus

By the Hirzebruch-Riemann-Roch theorem the index of the Dolbeault operator is the Todd genus.

Relation to Spin cSpin^c-structures

A complex manifold, being in particular an almost complex manifold, carries a canonical spin^c structure. The corresponding Spin^c Dirac operator identifies with the Dolbeault operator under the identification of the spinor bundle with that of holomorphic differential forms

S(X) 0,T *X. S(X) \simeq \wedge^{0,\bullet} T^\ast X \,.

References

  • Claire Voisin, section 2.3 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

  • Zachary Maddock, Dolbeault cohomology (pdf)

  • Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs, N.J., (1965)

  • Jean-Pierre Serre, Quelques problèmes globaux relatifs aux variétés de Stein, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Georges Thone, Liège, 1953, pp. 57–68. MR 0064155 (16,235b)

A formal geometry version:

Revised on June 4, 2014 22:19:16 by Urs Schreiber (82.136.246.44)