## Recent developments in the theory of separately holomorphic mappings.(English)Zbl 1189.32005

F. Hartogs discovered that functions of n variables which are holomorphic in each variable separately, are, in fact, holomorphic. This striking result has given rise, in recent years, to a host of more general theorems of a similar kind: one considers maps on product manifolds $$X$$ times $$Y$$ separately holomorphic on each factor, with values in some complex space $$Z$$, and asks whether they are holomorphic on the product, and one still modifies and weakens the assumptions in various ways. Pioneering work in this direction was done by J. Siciak and V. Zahariuta in the early 70s; their results, problems and methods have then been taken up and further developed by many people - there is still intensive research going on.
The author - who is one of the main contributors to the field - gives a unified and well structured account of the actual state of the art.
Reviewer: Ingo Lieb (Bonn)

### MSC:

 32D15 Continuation of analytic objects in several complex variables 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 32D10 Envelopes of holomorphy 32Uxx Pluripotential theory

### Keywords:

separate holomorphy
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