C*-invariante elliptische Faserungen

Abstract

Let X be a two-dimensional and Y a one-dimensional complex manifold. Consider a proper, connected, surjective holomorphic mapping p: X->Y. Assume that the inverse images of regular values of p (the regular fibers) are elliptic curves. Consider further a holomorphic action of the group of units C* of the complex numbers C on X such that p is invariant. Then p: X->Y is a C*-invariant elliptic fibration. Glas and Hausen showed that there are exactly two types of singular fibers: multiple elliptic curves (in Kodaira's classification fibers of type mI0 with m>1) and cycles of rational curves with or without multiplicity (fibers of type mIb with m>=1 and b>=1). First, we recall the theory of holomorphic C*-actions on complex lines and surfaces and the results of Glas and Hausen on C*-invariant elliptic fibrations with critical fibers with multiplicity one. Then we approach the open question of multiple critical fibers. We construct a C*-invariant elliptic fibration with a critical fiber of type mI0. Under a technical condition on the isotropy of the critical fiber, this modelclassifies all C*-invariant elliptic fibrations in the neighborhood of a mI0-type fiber up to equivariant biholomorphy. In a separate chapter, we give a short introduction to the theory of toric varieties. We then use toric varieties to generalize a local model of a mIb-type fiber due to Glas and Hausen to the case of multiplicities. inally, we give some more results of Glas and Hausen. If all singular fibers of a C*-invariant elliptic fibration are of multiplicity one, the fibration can be globally classified up to equivariant biholomorph.