Plurigenera of compact connected strongly pseudoconvex CR manifolds

Strongly pseudoconvex CR manifolds are boundaries of Stein varieties with isolated normal singularities. We introduce a series of new invariant plurigenera δm,m ∈ ℤ+ for a strongly pseudoconvex CR manifold. The main purpose of this paper is to present the following result: Let X1 and X2 be two compact strongly pseudoconvex embeddable CR manifolds of dimension 2n−1 ⩾ 3. If there is a non-constant CR morphism from X1 to X2, then δm(X2) ⩽ δm(X1) where δm(Xi) is the plurigeneus of Xi (see Definition 2.4).