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 2_n-1 ≥ 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 2_n – _N_2 – 1 ≥ 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.