nLab
Dolbeault complex

Contents

Context

Complex geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          The Dolbeault 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 \wedge 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 Dolbeault 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:

          Last revised on November 30, 2015 at 05:09:25. See the history of this page for a list of all contributions to it.