The Dolbeault Isomorphism

Abstract

In proving Mittag-Leffler’s theorem $\left( {{H^1}\left( {U,O} \right) = 0,U \subset {\Bbb C}} \right)$, we reduced the result to solving the equation $\frac{{\partial u}}{{\partial \bar z}} = f.$ The method given there, when formalised, leads to an important interpretation of H ¹ (X, E) [E being a holomorphic vector bundle on the Riemann surface X] called the Dolbeault isomorphism.