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We present the envelope of holomorphy of a classical truncated tube domain.

We discuss some basic properties of the Sibony functions and pseudometrics.

We present an effective formula for the Sibony function for all Reinhardt domains.

We continue to discuss the example presented in \cite{JarPfl2015}. In particular, we clarify some gaps and complete the description of the Shilov boundary.

We give a counterexample to the following theorem of Bremermann on Shilov boundaries: if $D$ is a bounded domain in $\mathbb C^n$ having a univalent envelope of holomorphy, say $\widetilde D$, then the Shilov boundary of $D$ with respect to the algebra $\mathcal A(D)$ coincides with the corresponding one for $\widetilde D$.

Let $p:X\longrightarrow M$ be a Riemann domain over a connected $n$-dimensional complex submanifold $M$ of $\mathbb C^N$ and let $\mathcal F\subset\mathcal O(X)$ be such that $p\in\mathcal F^N$. Our aim is to discuss relations between the $\mathcal F$-envelope of holomorphy of $(X,p)$ in the sense of Riemann domains over $M$ and the $\mathcal F$-envelope of holomorphy of $X$ in the sense of complex manifolds.

We prove a new cross theorem for separately holomorphic functions.

We prove that in the extension theorem for separately holomorphic functions on an $N$-fold cross with singularities the case of analytic singularities follows from the case of pluripolar singularities.

We present an elementary proof of the cross theorem in the case of Reinhardt domains. The results illustrates the well-known interrelations between the holomorphic geometry of a Reinhardt domain and the convex geometry of its logarithmic image., Comment: 7 pages

Let $X$ be a Riemann domain over $\mathbb C^k\times\mathbb C^\ell$. If $X$ is domain of holomorphy with respect to a family $\mathcal F\subset\mathcal O(X)$, then there exists a pluripolar set $P\subset\mathcal C^k$ such that every slice $X_a$ of $X$ with $a\notin P$ is a domain of holomorphy with respect to the family $\{f|_{X_a}: f\in\mathcal F\}$., Comment: 7 pages

We characterize the group $\operatorname{Aut}(\mathbb G_2)$ for the symmetrized bidisc $\mathbb G_2:=\{(\lambda_1+\lambda_2,\lambda_1\lambda_2):|\lambda_1|,|\lambda_2|<1\}\sub set\mathbb C^2$., Comment: 3 pages

We discuss continuity and upper semicontinuity of the Wu pseudometric., Comment: 5 pages

We prove that the category of generalized holomorphically contractible families possesses maximal and minimal objects. Moreover, we present basic properties of these extremal families., Comment: 15 pages

Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset X$ be relatively closed. For any $j\in\{1,...,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1})\times(A_{j+1}\times...\times A_N)$ such that the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb C^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,...,\Sigma_N$ are pluripolar. Put ${multline*} X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times...\times A_{j-1})\times D_j \times(A_{j+1}\times...\times A_N): (z',z'')\notin\Sigma_j\}$. Then there exists a relatively closed pluripolar subset $\widetilde M\subset\widetilde X$ of the `envelope of holomorphy' $\widetilde X$ of $X$ such that: $\bullet$ $\widetilde M\cap X'\subset M$, $\bullet$ every function $f$ separately meromorphic on $X\setminus M$ extends to a (uniquely determined) function $\widetilde f$ meromorphic on $\widetilde X\setminus\widetilde M$, $\bullet$ if $f$ is separately holomorphic on $X\setminus M$, then $\widetilde f$ is holomorphic on $\widetilde X\setminus\widetilde M$, and $\bullet$ $\widetilde M$ is singular with respect to the family of all functions $\widetilde f$. \noindent In the case where N=2, $M=\varnothing$, the above result may be strengthened., Comment: 11 pages

Let $D_j\subset\Bbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset\Bbb C^{n_1}\times...\times\Bbb C^{n_N}=\Bbb C^n. $$ Let $U\subset\Bbb C^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,...,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1}) \times(A_{j+1}\times...\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\Bbb C^{n_j}\: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,...,\Sigma_N$ are pluripolar. Put $$ X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times...\times A_{j-1})\times D_j \times(A_{j+1}\times...\times A_N)\: (z',z'')\notin\Sigma_j\}. $$ Then there exists a relatively closed pluripolar subset $\hat M\subset\hat X$ of the `envelope of holomorphy' $\hat X\subset\Bbb C^n$ of $X$ such that: $\hat M\cap X'\subset M$, for every function $f$ separately holomorphic on $X\setminus M$ there exists exactly one function $\hat f$ holomorphic on $\hat X\setminus\hat M$ with $\hat f=f$ on $X'\setminus M$, and $\hat M$ is singular with respect to the family of all functions $\hat f$. Some special cases were previously studied in \cite{Jar-Pfl 2001c}., Comment: 19 pages

This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089., Comment: This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb C^{k_1+...+k_N}.$$Let $U$ be an open connected neighborhood of $X$ and let $M\varsubsetneq U$ be an analytic subset. Then there exists an analytic subset $\hat M$ of the `envelope of holomorphy' $\hat X$ of $X$ with $\hat M\cap X\subset M$ such that for every function $f$ separately holomorphic on $X\setminus M$ there exists an $\hat f$ holomorphic on $\hat X\setminus\hat M$ with $\hat f|_{X\setminus M}=f$. The result generalizes special cases which were studied in \cite{\"Okt 1998}, \cite{\"Okt 1999}, \cite{Sic 2000}, and \cite{Jar-Pfl 2001}., Comment: 20 pages; This a new version of the paper (including "An extension theorem for separately holomorphic functions with singularities, II")

Let $D, G\subset{\Bbb C}$ be domains, let $A\subset D$, $B\subset G$ be locally regular sets, and let $X:=(D\times B)\cup(A\times G)$. Assume that $A$ is a Borel set. Let $M$ be a proper analytic subset of an open neighborhood of $X$. Then there exists a pure 1-dimensional analytic subset $\hat M$ of the envelope of holomorphy $\hat X$ of $X$ such that any function separately holomorphic on $X\setminus M$ extends to a holomorphic function on $\hat X\setminus\hat M$. The result generalizes special cases which were studied in \cite{\"Okt 1998}, \cite{\"Okt 1999a}, and \cite{Sic 2000}., Comment: 8 pages

We present various characterizations of $n$-circled domains of holomorphy $G\subset\CC^n$ with respect to some subspaces of $\Cal H^\infty(G)$.