13 - Euler characteristics and a typical values

Summary

Abstract

The theorem of Hà and Lê says that one can check using the Euler characteristic of the fibres whether a polynomial mapping ℂ² → ℂ is locally trivial in the sense that it defines a C^∞ fibre bundle. This theorem will be generalized to the case of a polynomial mapping g : Z → ℂ, where Z is a smooth closed algebraic subvariety of some ℂ^N, not necessarily of dimension 2. It is well-known that even in the case Z = ℂⁿ, n ≥ 3, it is no longer enough to look at the Euler characteristic of the fibre of g alone without serious additional assumptions. In this paper we will use the Euler characteristic of other spaces for this purpose in order to avoid explicit reference to some compactification.

The method of proof is the following: first it is shown that there is no vanishing cycle at infinity (with respect to a suitable compactification) and then that one can construct vector fields which lead to a local trivialization.

Introduction. Let g : ℂⁿ → ℂ be a polynomial map. It is well-known that even if g has no critical points it may happen that g does not define a fibre bundle which is locally (and therefore globally) C^∞ trivial. This is due to the circumstance that g is not proper as soon as n > 1, so we cannot apply Ehresmann's theorem.