Dolbeault cohomology

Summary

Dolbeault cohomology is the cohomology of the complex of smooth (p,q)-forms on a complex manifold with the dbar operator as differential. It measures to what extent the equation $\bar\partial u = f$ fails to be solvable for $\bar\partial$-closed data $f$. By the Dolbeault isomorphism it computes the sheaf cohomology of the sheaves of holomorphic $p$-forms. On compact Kähler manifolds it gives the Hodge decomposition of de Rham cohomology.