The local image problem for complex analytic maps

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 $(\mathbb{C}^2, 0)$. 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.

Keywords

set germs, analytic sets, analytic maps

2010 Mathematics Subject Classification

32B10, 32C25, 32H02, 32H35

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The authors acknowledge the support of the Labex CEMPI (ANR-11-LABX-0007-01). The first author acknowledges the CNCS grant PN-III-P4-ID-PCE-2016-0330.

Received 3 March 2021

Received revised 27 May 2021

Accepted 7 June 2021

Published 11 November 2021