Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds

Abstract

Let $X_1$ and $X_2$ be two compact strongly pseudoconvex CR manifolds of dimension $2n-1\ge 5$ which bound complex varieties $V_1$ and $V_2$ with only isolated normal singularities in $C^{N_1}$ and $C^{N_2}$ respectively. Let $S_1$ and $S_2$ be the singular sets of $V_1$ and $V_2$ respectively and $S_2$ is nonempty. If $2n–N_2–1\ge 1$ and the cardinality of $S_1$ is less than $2$ times the cardinality of $S_2$, then we prove that any non-constant CR morphism from $X_1$ to $X_2$ is necessarily a CR biholomorphism. On the other hand, let $X$ be a compact strongly pseudoconvex CR manifold of dimension $3$ which bounds a complex variety $V$ with only isolated normal non-quotient singularities. Assume that the singular set of $V$ is nonempty. Then we prove that any non-constant CR morphism from $X$ to $X$ is necessarily a CR biholomorphism.