Arkiv för Matematik

Volume 59 (2021)

Number 2

The local image problem for complex analytic maps

Pages: 345 – 358

DOI: https://dx.doi.org/10.4310/ARKIV.2021.v59.n2.a5

Authors

Cezar Joiţa (Institute of Mathematics of the Romanian Academy, Bucharest, Romania)

Mihai Tibăr (Laboratoire Paul Painlevé, Université de Lille, France)

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

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