Locally trivial fibrations with singular 1-dimensional Stein fiber over q-complete spaces

Abstract

Abstract. In connection with Serre's problem, we consider a locally trivial analytic fibration π : E −→ B of complex spaces with typical fiber X. We show that if X is a Stein curve and B is q-complete, then E is q-complete. §1. Introduction Let π : E → B be a locally trivial analytic fibration of complex spaces with Stein fiber X of dimension n.The following question was raised by Serre [17]:Under the above assumptions, does it follow that E is Stein if B is Stein?The answer is 'Yes' for n = 0 Related to this circle of ideas we study the case when the base B is q-complete. The normalization is chosen such that Stein spaces correspond to 1-complete spaces.For n = 0, i.e., E is a topological covering of B, Ballico [2] proved the q-completeness of E. This is a particular case of a result due to Vâjâitu [24] which gives that if π : Y → Z is a locally trivial analytic fibration with hyperconvex fibre and Z is q-complete, then Y is q-complete. (A complex space S is said to be hyperconvex if S is Stein and has a negative exhaustion function which is continuous and plurisubharmonic.)For n = 1, in order to generalize Mok's result, Vâjâitu [26] showed if X is non-singular, E is q-complete if B is q-complete. It remained the open problem when X is a singular Stein curve.

$q$-convexity properties of locally semi-proper morphisms of complex spaces

“…Colt ¸oiu and Vâjâitu [5] proved that if π : E → B is a locally analytic fibration of complex spaces such that the fiber is a Stein curve and B is q-complete, then E is q-complete. The case when E is a topological covering of B was already done in [3].…”

“…In [5], Colt ¸oiu and Vâjâitu considered locally trivial analytic fibrations π : E → B such that the fiber is a Stein curve and B is q-complete. In this way they improved the result of [3].…”