Global embeddings of weakly pseudoconvex complex spaces and refined Runge-type approximation theorems

Abstract

Runge-type approximation principles for holomorphic sections of adjoint line bundles are known only for weakly pseudoconvex manifolds. In this paper, we establish a refined form of such principles adapted to the setting of complex spaces and show that they yield global holomorphically embeddings for the regular locus of weakly pseudoconvex complex spaces. The key point is the construction of sequences of singular Hermitian metrics after a canonical resolution of singularities, together with a control of multiplier ideal sheaves via the strong openness property. This refined Runge-type approximation principle enables the globalization of local sections even when singularities persist at infinity. As an application, we solve the Union problem for weakly pseudoconvex complex manifolds.

v3: 21 pages; added Theorem 1.5 and made minor revisions. v2: 19 pages; added Examples 2.11, 2.12 and 2.13, Theorem 5.4, and Corollary 5.5. v1: 19 pages