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We describe a part of the recent developments in the theory of separately holomorphic mappings between complex analytic spaces. Our description focuses on works using the technique of holomorphic discs.

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f:W:=(A\times Y)\cup (X\times B)\to Z$ extends to a holomorphic mapping $\widehat{f}$ on $\widehat{W}:=\left\{(z,w)\in X\times Y:\tilde{\omega }(z,A,X)+\tilde{\omega }(w,B,Y)\<1\right\}$ such that $\widehat{f}=f$ on $W\cap \widehat{W},$ where $\tilde{\omega }(·,A,X)$ (resp. $\tilde{\omega }(·,B,Y))$ is the plurisubharmonic measure of $A$ (resp. $B$) relative to $X$ (resp. $Y$). Generalizations of...

We extend the theory of separately holomorphic mappings between complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay theorem on holomorphic discs and our recent joint-work with Pflug on boundary cross theorems in dimension $1.$ It also relies on our new technique of conformal mappings and a generalization of Siciak’s relative extremal function. Our approach illustrates the unified character: “From local information to global extensions”. Moreover, it avoids systematically...

We first introduce the class of quasi-algebraically stable meromorphic maps of P. This class is strictly larger than that of algebraically stable meromorphic self-maps of P. Then we prove that all maps in the new class enjoy a recurrent property. In particular, the algebraic degrees for iterates of these maps can be computed and their first dynamical degrees are always algebraic integers.

Let D ⊂ ℂⁿ and $G\subset {ℂ}^m$ be pseudoconvex domains, let A (resp. B) be an open subset of the boundary ∂D (resp. ∂G) and let X be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Suppose in addition that the domain D (resp. G) is locally ² smooth on A (resp. B). We shall determine the “envelope of holomorphy” X̂ of X in the sense that any function continuous on X and separately holomorphic on (A×G)∪(D×B) extends to a function continuous on X̂ and holomorphic on the interior of X̂. A generalization of this result to N-fold...

Let X, Y be two complex manifolds of dimension 1 which are countable at infinity, let D ⊂ X, G ⊂ Y be two open sets, let A (resp. B) be a subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D∪A)×B) ∪ (A×(B∪G)). Suppose in addition that D (resp. G) is Jordan-curve-like on A (resp. B) and that A and B are of positive length. We determine the "envelope of holomorphy" Ŵ of W in the sense that any function locally bounded on W, measurable on A × B, and separately holomorphic on (A × G) ∪ (D × B)...

We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions.

In this paper we develop the H(p ≥ 1) theory on the minimal ball. After identifying the admissible approach regions, we establish theorems of Fatou and Koráanyi-Vági type on this ball.