 Solution

Let ${\mathcal{A}}^{p,q}(X)$ be the space of smooth complex differential forms of type $(p,q)$. The Dolbeault operator satisfies ${\bar{\partial }}^2=0$, and the Dolbeault cohomology isFor $q=0$ the denominator is zero; forms outside the dimension range are zero.

A sequence of sheaves is an exact sequence of sheaves precisely when its sequence of stalks at every point is exact. Here this means the first map is injective on stalks, the last is surjective on stalks, and the image equals the kernel at the middle stalk. In particular, a section of the last sheaf need only have local lifts; surjectivity on all global sections is not required.

The associated long exact sequence in sheaf cohomology of Čech cohomology, understood in the direct limit over open covers, isThe degree-zero groups are global sections. The connecting map is obtained by locally lifting a Čech cocycle to the middle sheaf and taking its Čech coboundary, which takes values in the first sheaf. Different lifts change it by a Čech coboundary.