Convergence and divergence of formal CR mappings

Let $M \subset \mathbb{C}^N$ be a generic real-analytic submanifold of finite type, $M' \subset \mathbb{C}^{N'}$ be a real-analytic set, and $p \in M$, where we assume that $N, N' \geqslant 2$. Let $H: (\mathbb{C}^N, p) \to \mathbb{C}^{N'}$ be a formal holomorphic mapping sending $M$ into $M'$, and let $\mathcal{E}_{M'}$ denote the set of points in $M'$ through which there passes a complex-analytic subvariety of positive dimension contained in $M'$. We show that, if $H$ does not send $M$ into $\mathcal{E}_{M'}$, then $H$ must be convergent. As a consequence, we derive the convergence of all formal holomorphic mappings when $M'$ does not contain any complex-analytic subvariety of positive dimension, answering by this a long-standing open question in the field. More generally, we establish necessary conditions for the existence of divergent formal maps, even when the target real-analytic set is foliated by complex-analytic subvarieties, allowing us to settle additional convergence problems such as e.g. for transversal formal maps between Levi-non-degenerate hypersurfaces and for formal maps with range in the tube over the light cone.

Keywords

formal CR map, convergence, deformation, complex-analytic subvariety

2010 Mathematics Subject Classification

32H02, 32H40, 32V20, 32V25, 32V40

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The authors were partially supported by the Qatar National Research Fund, NPRP project 7-511-1-098. The first author was also supported by the Austrian Science Fund FWF, Project I1776.

Received 26 September 2017

Accepted 18 July 2018

Published 16 August 2018