The local image problem for complex analytic maps

TL;DR

This paper establishes a criterion for when holomorphic map germs are locally open, resolving a conjecture by Huckleberry across all dimensions, and clarifies the conditions under which the local image of complex analytic maps is well defined.

Contribution

It provides a comprehensive criterion for local openness of holomorphic map germs, solving Huckleberry's conjecture in all dimensions.

Findings

Proves a criterion for local openness of holomorphic map germs.

Addresses the local image problem for complex analytic maps.

Solves Huckleberry's conjecture in all dimensions.

Abstract

We address the question "when the local image of a map is well defined" and answer it in case of holomorphic map germs with target $(\bC^{2}, 0)$. We prove a criterion for holomorphic map germs $(X, x)\to (Y, y)$ to be locally open, solving a conjecture by Huckleberry in all dimensions.