Deformations of CR maps and applications

We study the deformation theory of CR maps in the positive codimensional case. In particular, we study structural properties of the mapping locus $E$ of (germs of nondegenerate) holomorphic maps $H: (M, p) \to M^\prime$ between generic real submanifolds $M \subset \mathbb{C}^N$ and $ M^\prime \subset \mathbb{C}^{N^\prime}$, defined to be the set of points $p^\prime \in M^\prime$ which admit such a map with $H(p) = p^\prime$. We show that this set $E$ is semi-analytic and provide examples for which E possesses (prescribed) singularities.

Keywords

CR maps, deformations of CR manifolds, mapping locus, jet parametrization property, semi-analytic sets

2010 Mathematics Subject Classification

32H02, 32V40

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G.D.S. was partially supported by the Center for Advanced Mathematical Sciences and by the grant “Complex Geometry of Real Manifolds” by the University Research Board of AUB.

B.L. was partially supported by the Austrian Science Fund (FWF) project I3472 and I4557.

M.R. was supported by the Austrian Science Fund (FWF) project P28873.

Received 15 November 2018

Accepted 25 March 2021

Published 6 July 2022