Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds (Q610659)

Summary: Let \(X_{1}\) and \(X_{2}\) be two compact strongly pseudoconvex CR manifolds of dimension \(2n-1 \geq 5\) which bound complex varieties \(V_{1}\) and \(V_{2}\) with only isolated normal singularities in \(\mathbb C^{N1}\) and \(\mathbb C^{N2}\) respectively. Let \(S_{1}\) and \(S_{2}\) be the singular sets of \(V_{1}\) and \(V_{2}\) respectively and \(S_{2}\) is nonempty. If \(2n - N_{2} - 1 \geq 1\) 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.