Recent developments in the theory of separately holomorphic mappings

Abstract

Abstract. 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.

Conical plurisubharmonic measure and new cross theorems

“…Theorem 3.5) is a particular case of Theorem 10.4 (resp. Theorem 10.5) in [4]. However, the proofs of the latter theorems are analogous to those of the former ones.…”

“…The purpose of this section is to derive from Theorem A in our previous work [4] and Theorem 1.1 a boundary cross theorem and a mixed cross theorem. We first recall some terminology and notation (relative to the system of conical approach regions) introduced in Section 2 of [4]. Moreover, A is said to be locally pluriregular if it is locally pluriregular at all points a ∈ A.…”

Vyacheslav Zakharyuta’s complex analysis

“…In subsequent studies, the extremal function V E as well as the relative extremal function have played important roles in the proofs of generalizations of this theorem of Hartogs; see, e.g., Siciak (1969), Nguyen Thanh Van andZeriahi (1983), Shiffman (1989) and Nguyên (2008). The last-mentioned paper contains new results as well as a careful study of the history of the subject.…”

A unified approach to the theory of separately holomorphic mappings

“…On the other hand, in [37] the problem of optimality of the envelope of holomorphy W in Theorem A has been investigated. A survey on recent developments in the theory of separately holomorphic mappings could be found in [30].…”