L2∂¯-cohomology with weights and bundle convexity of certain locally pseudoconvex domains

Abstract

Employing a variant of Hörmander’s approach to Andreotti–Grauert’s theory, a comparison theorem is proved between some bundle-valued weighted $L^2$ $\overline{\partial }$-cohomology groups of a class of locally pseudoconvex bounded domains in complex manifolds. As a result, such domains will turn out to be domains of meromorphy if the manifold admits a Hermitian line bundle whose curvature form is positive on the boundary of the domain. Particularly, a domain in a compact complex algebraic surface admits a meromorphic function whose essential singularities are the points on the boundary if the complement of the domain is a complex curve of self-intersection zero.