TITLE: Why only consider Dolbeault cohomology?
QUESTION [14 upvotes]: On a complex manifold we have the differential operators
$$\partial:A^{p,q}\to A^{p+1,q}$$
$$\bar\partial:A^{p,q}\to A^{p,q+1}$$
which both square to zero. Hence one can define cohomology groups
$$H^{p,q}_\partial=\frac{\ker\partial:A^{p,q}\to A^{p+1,q}}{\partial A^{p-1,q}}$$
$$H^{p,q}_{\bar\partial}=\frac{\ker\bar\partial:A^{p,q}\to A^{p,q+1}}{\bar\partial A^{p,q-1}}$$
But for some reason I only ever see people discussing the $H_{\bar\partial}^{p,q}$, i.e. Dolbeaut cohomology. What makes this more interesting the the $H_\partial^{p,q}$ groups?

REPLY [14 votes]: $\newcommand{\dd}{\partial}$To supplement the existing (excellent) answers: If $E \to M$ is a holomorphic vector bundle, the transition functions of $E$ are "constant with respect to $\bar{\dd}$", so there's a natural notion of $E$-valued Dolbeault cohomology $H_{\bar{\dd}}^{p,q}(M, E)$. Unless the transition functions of $E$ are constant, however, there is no corresponding notion of $H_{\dd}^{p,q}(M, E)$.