Steinness of the Total Space of a Non-Trivial Algebraic Affine ℂ-Bundle on the Punctured Complex Affine Plane
✍ Scribed by Makoto Abe
- Publisher
- John Wiley and Sons
- Year
- 2002
- Tongue
- English
- Weight
- 161 KB
- Volume
- 238
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
We prove that the total space E of an algebraic affine C -bundle π : E → X on the punctured complex affine plane X := C 2 -{(0, 0)} is Stein if and only if it is not isomorphic to the trivial holomorphic line bundle X × C .
0. Introduction
Let π : E → X be a holomorphic affine C -bundle on the punctured complex affine plane
can be chosen such that both of the functions a and b are meromorphic on C 2 , then E is said to be algebraic (cf. Proposition 1.2). The special linear group SL(2, C ) is the simplest example of a Stein manifold which is realized as an algebraic affine C -bundle on X = C 2 -{(0, 0)} (cf. [1], p. 545).
In this paper we prove that the total space E of an algebraic affine C -bundle on X = C 2 -{(0, 0)} is Stein if and only if the bundle E is not isomorphic to the trivial holomorphic line bundle X × C (Theorem 3.1). For the proof of this fact we need the author's result on the Steinness of the total space of a quasi-trivial holomorphic affine C -bundle on the complement of an analytic set of codimension 2 in a normal Stein space (Theorem of [1]).