Holomorphic extensions in complex fiber bundles

Abstract

The main result of this paper is the existence of solutions of the equation ∂ ¯ u = ω for closed ( 0 , 1 ) -forms with compact support in a complex fiber bundle from some class. This class of bundles contains, for example, vector bundles. What is important, the solution exists for any base manifold. The solvability property implies many results about extension and representation of complex analytic and Cauchy–Riemann functions: Riemann–Hilbert type representation, Hartogs and Hartogs–Bochner phenomena, vanishing of some cohomology groups, to mention few.