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 ${ℂ}^{N1}$ and ${ℂ}^{N2}$ respectively. Let $S_1$ and $S_2$ be the singular sets of $V_1$ and $V_2$ respectively and $S_2$ is nonempty. If $2n-N_2-1\ge 1$ 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...

We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain $D_{n,m}^p\left(\mu \right)$. The generalized Fock-Bargmann-Hartogs domain is defined by inequality ${\mathrm{e}}^{{\mu \parallel z\parallel }^2}{\sum }_{j=1}^m{\left|{\omega }_j\right|}^{2p}<1$, where $(z,\omega )\in {ℂ}^n\times {ℂ}^m$. In this paper, we will establish a rigidity of its holomorphic automorphism group. Our results imply that a holomorphic self-mapping of the generalized Fock-Bargmann-Hartogs domain $D_{n,m}^p\left(\mu \right)$ becomes a holomorphic automorphism if and only if it keeps the function ${\sum }_{j=1}^m{\left|{\omega }_j\right|}^{2p}{\mathrm{e}}^{{\mu \parallel z\parallel }^2}$ invariant.

Given a compact set $K\subset {ℂ}^N$, for each positive integer n, let $V^{\left(n\right)}\left(z\right)$ = $V_K^{\left(n\right)}\left(z\right)$ := sup$1/\left(degp\right)V_{p\left(K\right)}\left(p\left(z\right)\right)$: p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V_K^{\left(n\right)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_K\left(z\right)$:= max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${\left|\right|p\left|\right|}_K\le 1$]. Our main result is that if K is regular, then all of the functions $V_K^{\left(n\right)}$ are continuous; and their associated Robin functions ${\varrho }_{V_K^{\left(n\right)}}\left(z\right):=limsu{p}_{\left|\lambda \right|\to \infty }[V_K^{\left(n\right)}\left(\lambda z\right)-log\left(\right|\lambda \left|\right)]$ increase to ${\varrho }_K:={\varrho }_{V_K}$ for all z outside a pluripolar set....

The general Stein union problem is solved: given an increasing sequence of Stein open sets, it is shown that the union $X$ is Stein if and only if $H^1(X,{\mathbf{O}}_X)$ is Hausdorff separated.

A necessary and sufficient condition, which is a weak converse of a classical theorem of Behnke-Stein, in order that a limit of Stein spaces be again a Stein space is proved.