The Serre problem on certain bounded domains
The Serre problem consists of the following question: Given a holomorphic fiber bundle \(\pi:E\rightarrow B\) with a basis \(B\) and a fiber \(F\). Let \(F\) and \(B\) be Stein. Does it follow that also \(E\) is Stein? If the dimension of the fiber is \(0\) or \(1\) then this is known to be true. But in general the Serre problem does not admit a positive answer, as is shown by counterexamples that have been found by Skoda in 1977 and also Coeure-Loeb in 1985. In 1973 J.-L. Stehlé proved, that the Serre problem can be answered in a positive way, if the fiber satisfies the following condition: On \(F\) there exists a plurisubharmonic exhaustion function \(\psi\) such that \(\psi \circ h -\psi\) is bounded for any automorphism \(h\) of \(F\). In the article under review a criterion is given that is in the spirit of Stehlé's result: Theorem: The answer to the Serre problem is positive, if the fiber is either of the following: (i) a bounded domain \(\Omega \) in \(\mathbb{C}^n\) which has a plurisubharmonic exhaustion function \(\psi\) such that \(\psi \leq c\, \log \log (1/\delta_\Omega ) \), where \(\delta_\Omega\) denotes the boundary distance function on \(\Omega\) and \(c>0\) is a constant; (ii) a Stein domain \(\widetilde \Omega \setminus S\), where \(\widetilde \Omega\) is a bounded domain which has a continuous bounded plurisubharmonic exhaustion function \(\psi\) such that \(\psi \leq c \delta_\Omega ^\gamma \), where \(c,\gamma >0\), and \(S\) is a closed subset of \(\overline \Omega\) which is negligible w.r.t. holomorphic \(L^2\) functions, i.e. any holomorphic \(L^2\)-function on \(\Omega\) extends holomorphically to \(\widetilde \Omega\). Furthermore it is shown that a bounded hyperconvex domain always satisfies condition (i) and an example of a domain is given, that satisfies (i) without being hyperconvex.
scientific article; zbMATH DE number 2113425