A class of counter-examples to the hypersection problem based on forcing equations. (English) Zbl 1059.14049

Brenner, Holger

Arch. Math. 82, No. 6, 564-569 (2004).

From the introduction: Let \(W\) be a complex Stein space of dimension \(\geq 3\) and let \(H\subset W\) be an analytic hypersurface, \(U=W-H\). Suppose that for every analytic hypersurface \(S\subset W\) the intersection \(U\cap S\) is Stein, is then \(U\) itself Stein? This question is called the hypersection problem [see K. Diederich, in: Geometric complex analysis, 163–181 (1996; Zbl 0930.32019) for a general treatment and related problems]. The first counter-example to this question was given by M. Coltoiu and K. Diederich [Ann. Math. 145, 175–182 (1997; Zbl 0871.32008)], using the affine cone over the complement of two sections on some ruled surface over an elliptic curve. In this way a normal three-dimensional isolated singularity was constructed.
In this paper we present another class of three-dimensional Stein spaces \(W\) together with a hypersurface \(H\) fulfilling the assumptions in the hypersection problem, but not its conclusion. The class is constructed in the following way: we start with a two-dimensional normal affine cone \(X\) over a smooth projective curve and the vertex point \(P\in X\). Suppose that we have three homogeneous functions \(f_1\), \(f_2\) and \(f_0\) on \(X\). Then, under suitable conditions, \(W=V(f_1t_1+f_2t_2+f_0) \subset X\times \mathbb{C}^2\) and the hypersurface \(H=p^{-1} (P)\) have the desired properties.

MSC:

14J26	Rational and ruled surfaces
32C25	Analytic subsets and submanifolds
32E10	Stein spaces
32E40	The Levi problem
32Q28	Stein manifolds

Keywords:

complex Stein space; affine cone; ruled surface over an elliptic curve

Citations:

Zbl 0930.32019; Zbl 0871.32008