Differential Forms and Dolbeault Theory

Abstract

In this chapter Dolbeault cohomology theory is presented. One of the basic tools is the $\bar \partial$-integration lemma for closed (p, q)-forms (Theorem 4.1). The proof of this lemma is based on the existence of bounded solutions of the inhomogeneous Cauchy-Riemann differential equation $\partial g/\partial \bar z = f.$This solution is constructed in Paragraph 3 by means of the classical integral operator

$$Tf(z,u) = \frac{1}{{2\pi i}}\iint\limits_B {\frac{{f(\zeta ,u)}}{{\zeta - z}}}d\zeta \wedge d\bar \zeta .$$