Rigidity of CR morphisms

Xiankui Meng and Stephen Shing-Toung Yau

Vol. 316 (2022), No. 1, 207–216

DOI: 10.2140/pjm.2022.316.207

Abstract

Let $X_1$ and $X_2$ be two compact strongly pseudoconvex CR manifolds of dimension $2n-1\ge 5$ which are boundaries of complex varieties $V_1$ and $V_2$ with only isolated normal singularities in ${ℂ}^{N_1}$ and ${ℂ}^{N_2}$, respectively. We show that any nonconstant CR morphism from $X_1$ to $X_2$ can be extended to a finite holomorphic covering map from $V_1$ to $V_2$ in the case where $V_2$ has only isolated complete intersection singularities. In particular, there is no nonconstant CR morphism between the links of two isolated complete intersection singularities with different embedding codimensions. Finally, using Kohn–Rossi cohomology, we obtain some sufficient conditions for the nonexistence of CR morphisms from $X_1$ to $X_2$.

Dedicate to Professor H. Blaine Lawson on the occasion of his 80th birthday

Keywords

strongly pseudoconvex CR manifold, rigidity of CR morphism, isolated complete intersection singularity, Kohn–Rossi cohomology

Mathematical Subject Classification

Primary: 32S20, 32V99

Milestones

Received: 20 June 2021

Revised: 6 October 2021

Accepted: 5 November 2021

Published: 26 February 2022