Holomorphic fiber bundle with Stein base and Stein fibers
نویسنده
چکیده
A surjective holomorphic map Π : X → Y between complex spaces is said to be a Stein morphism, if for every point y ∈ Y , there exists an open neighborhood U of y such that Π(U) is Stein. In 1977, Skoda [12] showed that a locally trivial analytic fiber bundle Π : X → Ω with Stein base and Stein fibers is not necessarily a Stein manifold. Thus giving a negative answer to a conjecture proposed in 1953 by J-P. Serre [11]. In the counterexample provided by Skoda the base Ω is an open set in I C and the holomorphic functions on X are constant along the fibers Π(t), t ∈ Ω. It follows (cf. J-P. Demailly [4]) that the cohomology group H(X,OX) is an infinitely dimensional complex vector space. At the same time, Fornaess [6] solved by means of a 2-dimensional counterexample the analogous problem for the Stein morphism.
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