Dolbeault complex - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold of complex dimension $n$ , and fix $0\leq p\leq n$ . The Dolbeault complex in holomorphic degree $p$ is the cochain complex

0\longrightarrow\Omega^{p,0}(X)\xrightarrow{\bar\partial} \Omega^{p,1}(X)\xrightarrow{\bar\partial}\cdots \xrightarrow{\bar\partial}\Omega^{p,n}(X)\longrightarrow0,

where $\Omega^{p,q}(X)$ is the complex vector space of smooth $(p,q)$ -forms and $\bar\partial$ is the $(0,1)$ Dolbeault operator. The identity $\bar\partial^2=0$ makes consecutive arrows compose to zero. Unless $p$ is specified, “the Dolbeault complex” may also mean the resulting family over all $p$ . Each vector space and arrow is complex-linear, and the cochain grading is the antiholomorphic form degree $q$ .

Local exactness and sheaves

The Dolbeault lemma says that every $\bar\partial$ -closed $(p,q)$ -form with $q>0$ is locally $\bar\partial$ -exact. Consequently, the sheaf version

0\longrightarrow\Omega_X^p\longrightarrow\mathcal A_X^{p,0} \xrightarrow{\bar\partial}\mathcal A_X^{p,1}\longrightarrow\cdots

is a fine resolution of the sheaf $\Omega_X^p$ of holomorphic $p$ -forms. This local statement and its resolution interpretation are developed in Voisin, §2.3.3.

Structure and examples

For $p=0$ , the kernel of the first arrow consists exactly of holomorphic functions. More generally, the degree-zero cocycles are holomorphic $p$ -forms. In complex dimension one, each fixed- $p$ complex has only the two potentially nonzero terms $\Omega^{p,0}(X)$ and $\Omega^{p,1}(X)$ .

Wedge product makes the direct sum over $p$ and $q$ into a bigraded differential algebra: if $\alpha$ has total degree $k$ , then

\bar\partial(\alpha\wedge\beta) =\bar\partial\alpha\wedge\beta+(-1)^k\alpha\wedge\bar\partial\beta.

Conventions and scope

Some authors reserve “Dolbeault complex” for the $p=0$ row and call the general object the $p$ th Dolbeault complex. The construction requires an integrable complex structure. On a general almost-complex manifold, the exterior derivative can have additional type components, so the displayed sequence need not be a complex.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. DOI record. Relevant: §2.3.3, the Dolbeault complex of a holomorphic bundle.
R. O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. DOI record. Relevant: Chapter II, §3, especially Theorem 3.17, the Dolbeault theorem.