Dolbeault operators - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold. For a form of type $(p,q)$ , the exterior derivative has only two type components. The Dolbeault operators are their projections:

\partial:\Omega^{p,q}(X)\to\Omega^{p+1,q}(X), \qquad \bar\partial:\Omega^{p,q}(X)\to\Omega^{p,q+1}(X), \qquad d=\partial+\bar\partial.

They are complex-linear first-order differential operators. Since $d^2=0$ and the three resulting bidegrees are distinct, they satisfy $\partial^2=0$ , $\bar\partial^2=0$ , and $\partial\bar\partial+\bar\partial\partial=0$ . Thus each operator is intrinsic to the complex structure and independent of the chosen holomorphic coordinates.

Local formulas

In holomorphic coordinates $z^1,\ldots,z^n$ , for a smooth function $f$ ,

\partial f=\sum_j\frac{\partial f}{\partial z^j}\,dz^j, \qquad \bar\partial f=\sum_j\frac{\partial f}{\partial\bar z^j}\,d\bar z^j.

The operators extend to forms by the graded Leibniz rule. A smooth complex-valued function is holomorphic exactly when $\bar\partial f=0$ .

Dolbeault complexes

For fixed $p$ , the identity $\bar\partial^2=0$ makes

\Omega^{p,0}(X)\xrightarrow{\bar\partial}\Omega^{p,1}(X) \xrightarrow{\bar\partial}\Omega^{p,2}(X)\longrightarrow\cdots

a cochain complex. Its cohomology is Dolbeault cohomology. The analogous $\partial$ -complex fixes $q$ , and complex conjugation interchanges the two operators.

Integrability and scope

On an almost-complex manifold, $d$ may have additional components of bidegrees $(2,-1)$ and $(-1,2)$ . Their disappearance is equivalent to integrability, so the two-term formula above uses the complex-manifold hypothesis Wells, Chapter I, §3.

References

R. O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. DOI record. Relevant: Chapter I, §3, the operators $\partial$ and $\bar\partial$ .
C. Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge University Press, 2002. DOI record. Relevant: §2.1, bidegrees and Dolbeault complexes.