Dolbeault cohomology ($H_{\bar{\partial }}^{p,q}(X)$)

Dolbeault cohomology is

The Dolbeault theorem identifies this group with first Čech cohomology of the trivial holomorphic line bundle. On the two standard affine charts, a holomorphic overlap function has a Laurent series. Its nonnegative powers extend to the chart at zero, and its negative powers extend to the chart at infinity. Every overlap cocycle is therefore a coboundary. Consequently the Dolbeault operator from functions to $(0,1)$-forms is surjective.

For a smooth $(0,1)$-form on the complex projective line, solve its Dolbeault operator equation on a disc about zero and a disc about infinity. The difference of the primitives is a holomorphic function on their annular overlap. Split its Laurent series into a nonnegative-power part extending to the finite disc and a negative-power part extending holomorphically across infinity. Subtract the first part from the finite primitive and add the second to the infinite primitive. The results agree and define a global primitive, without Hodge theory.

On a Stein manifold, every smooth $\bar{\partial }$-closed $(0,q)$-form with $q>0$ is globally $\bar{\partial }$-exact. The same holds componentwise for forms valued in a finite-dimensional constant vector space. Complex conjugation yields the corresponding global $\partial$ primitive for a closed $(1,0)$-form, as used in the complex potential reduction of anti-self-dual Yang-Mills.

For $X={\mathbb{P}}^1\times \mathbb{C}$, the two standard affine charts and their intersection form an acyclic cover for the sheaves of holomorphic forms. Their Čech cohomology gives $H_{\bar{\partial }}^{0,0}=\mathcal{O}(\mathbb{C})$ and $H_{\bar{\partial }}^{2,1}=\mathcal{O}(\mathbb{C})$, with all other groups for $p\in \{0,2\}$ zero. For the top-form sheaf, the transition $d(1/z)=-z^{-2}dz$ removes every Laurent series power except $z^{-1}$ from the first cohomology quotient; its coefficient may be an arbitrary entire function on the affine factor.

This is the cohomology of ${\mathcal{A}}^{p,\bullet }(X,E)$ with the bundle Dolbeault operator. The Dolbeault theorem identifies it with $H^q(X,{\Omega }_X^p\otimes \mathcal{O}(E))$. For compact Hermitian $X$, Dolbeault Hodge decomposition gives unique harmonic representatives.

Chern curvature defines $\alpha (E)=[\Theta ]\in H^1(M,{\Omega }_M^1\otimes \mathrm{End}E)$. If two metrics give connections differing by $a$, then ${\Theta }_1-{\Theta }_0={\bar{\partial }}_{\mathrm{End}E}a$. Thus this vector-bundle curvature class depends on the holomorphic bundle rather than the chosen metric.

In frames with $e^{(\beta )}=g_{\alpha \beta }{e}^{(\alpha )}$ and input-index connection matrices, the Chern connection has ${\theta }_{\beta }-g_{\alpha \beta }{\theta }_{\alpha }{g}_{\alpha \beta }^{-1}={\sigma }_{\alpha \beta }$. These holomorphic one-forms, interpreted in the beta frame, satisfy the transported Čech cocycle condition because $g_{\alpha \gamma }=g_{\beta \gamma }{g}_{\alpha \beta }$. Local primitives of curvature differ by this cocycle. A holomorphic frame change $e_{\alpha }^{\prime }=u_{\alpha }{e}_{\alpha }$ adds the Čech coboundary of $u_{\alpha }^{-1}\partial u_{\alpha }$, so the resulting curvature Dolbeault class is intrinsic.

Composition of endomorphisms and wedge product of differential forms give the class $[{\Theta }^k]\in H^k(M,{\Omega }_M^k\otimes \mathrm{End}E)$. Even total degree of vector-bundle curvature and the noncommutative telescoping identity give ${\Theta }_1^k-{\Theta }_0^k=\bar{\partial }{\sum }_{j=0}^{k-1}{\Theta }_1^{j}a{\Theta }_0^{k-1-j}$ whenever ${\Theta }_1-{\Theta }_0=\bar{\partial }a$. Thus all powers are metric independent; no commutativity of endomorphisms is assumed.

For a complex torus $X=V/\Lambda$ of complex dimension $g$, constant forms give a natural isomorphism $H_{\bar{\partial }}^{p,q}(X)\cong {\bigwedge }^p{V}^{\ast }\otimes {\bigwedge }^q\overline{V^{\ast }}$. A flat Kähler metric and the Dolbeault Hodge decomposition prove that these constant forms are precisely the harmonic representatives. Thus $h^{p,q}=\left(\genfrac{}{}{0ex}{}{g}{p}\right)\left(\genfrac{}{}{0ex}{}{g}{q}\right)$ for $0\le p,q\le g$, and zero otherwise.

For a complex manifold, Bott-Chern cohomology isIts classes are represented by $d$-closed forms of pure type. A natural map sends a Bott-Chern class to its Dolbeault cohomology class.

For a compact Kähler manifold, the natural map $H_{BC}^{p,q}\to H_{\bar{\partial }}^{p,q}$ is an isomorphism. The Kähler Laplacian identity makes the harmonic representative of every Dolbeault cohomology class $d$-closed, giving surjectivity, and the ddbar lemma gives injectivity.

On a polydisc, $H_{BC}^{p,q}=0$ for $p,q>0$. A Poincaré lemma primitive for a closed pure-type form can be modified by exact terms until only bidegrees $(p-1,q)$ and $(p,q-1)$ remain. The Dolbeault-Poincaré lemma and conjugate Dolbeault-Poincaré lemma then make the original form $\partial \bar{\partial }$-exact.

For the sheaf ${\Omega }^p$ of holomorphic $p$-forms on a complex manifold, the Dolbeault-Poincaré lemma makesa resolution by fine sheaves. Taking global sections therefore gives the natural isomorphism

The Dolbeault-Poincaré lemma makes this a locally exact complex of sheaves, with degree-zero kernel the holomorphic differential forms. The sheaves of smooth forms are fine sheaves by a partition of unity. The acyclic resolution theorem therefore computes the cohomology of ${\Omega }^p$ by the global Dolbeault operator complex.

The Hodge number of a compact complex manifold is $h^{p,q}={\dim }_{\mathbb{C}}{H}_{\bar{\partial }}^{p,q}(X)$.

For a compact Kähler manifold, these dimensions are the graded dimensions in its Hodge structure.

On a compact Kähler manifold of complex dimension $n$, the conjugate-linear Hodge star gives a bijection of harmonic forms in complementary bidegrees. The Dolbeault Hodge decomposition identifies these harmonic spaces with Dolbeault cohomology and yields the displayed equality.

On a compact Kähler manifold, complex conjugation commutes with the real Hodge Laplacian and exchanges harmonic forms of bidegrees $(p,q)$ and $(q,p)$. The identification of Dolbeault cohomology with harmonic forms therefore gives the symmetry of Hodge numbers.