Kohn–Rossi cohomology and nonexistence of CR morphisms between compact strongly pseudoconvex CR manifolds

Abstract

One of the fundamental questions in CR geometry is: Given two strongly pseudoconvex CR manifolds $X_1$ and $X_2$ of dimension $2n-1$, is there a non-constant CR morphism between them? In this paper, we use Kohn–Rossi cohomology to show the non-existence of non-constant CR morphism between such two CR manifolds. Specifically, if $\dim H^{p,q}_{KR} (X_1) \lt \dim H^{p,q}_{KR} (X_2)$ for any $(p, q)$ with $1 \leq q \leq n-2$, then there is no non-constant CR morphism from $X_1$ to $X_2$.