Euler characteristics and atypical values

摘要

The theorem of Ha and Le says that one can check using the Euler characteristic of the fibres whether a polynomial mapping C~2→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 = C~n, 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.