Approximation of holomorphic mappings on 1-convex domains

TL;DR

This paper develops approximation techniques for holomorphic sections over 1-convex domains, enabling better control and extension of such sections, with applications to constructing proper holomorphic maps into q-convex manifolds.

Contribution

It introduces a spray of -sections with prescribed core and approximation properties on 1-convex domains, extending the Oka principle in this setting.

Findings

Existence of sprays of -sections with prescribed core

Approximation of -sections by holomorphic sections in neighborhoods

Application to constructing proper holomorphic maps into q-convex manifolds

Abstract

Let \pi :Z\to X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D\Subset X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of \pi -sections over \bar{D} which has prescribed core, it fixes the exceptional set E of D, and is dominating on \bar{D}\setminus E. Each section in this spray is of class C^k(\bar{D}) and holomorphic on D. As a consequence we obtain several approximation results for \pi -sections. In particular, we prove that \pi -sections which are of class C^k(\bar{D}) and holomorphic on D can be approximated in the C^k(\bar{D}) topology by \pi -sections that are holomorphic in open neighborhoods of \bar{D}. Under additional assumptions on the submersion we also get approximation by global holomorphic \pi -sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.