Abstract
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.
Funding Statement
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.
Citation
Bernhard Lamel. Nordine Mir. "Convergence and divergence of formal CR mappings." Acta Math. 220 (2) 367 - 406, June 2018. https://doi.org/10.4310/ACTA.2018.v220.n2.a5
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