AN ADJUNCTION FORMULA FOR LOCAL COMPLETE INTERSECTIONS

Abstract

In this article, we study various kinds of indices of a vector field on a singular variety and as an application, we prove, for a compact strong local complete intersection V with isolated singularities, a formula expressing the Euler-Poincare characteristic x (^ ) 0 I ^ i n terms of the top Chern class of the tangent bundle of V and the Milnor numbers of the singularities (Theorem 2.4). For a vector field t i o n a singular variety V, we consider the index, the GSV-index and the virtual at the singularity of v. All these reduce to the usual Poincare-Hopf index when the singularity of v is in the regular part of V, so we compare them when it is in the singular part of V. M.-H. Schwartz defined an index for vector fields on a singular variety V, see [21, 4]. When the singularities of V are isolated, as they are in this article, this definition can be easily extended to vector fields which are not radial. We do this in Sec. 1 below and we call the corresponding index the Schwartz index of a vector field. We show that, for a global vector field with isolated singularities on a compact variety V, the sum of the Schwartz indices gives x(V) (Theorem 1.2). In [12] there is a definition of a local index for stratified vector fields on singular varieties, extending Schwartz' definition for radial vector fields. Presumably our definition of the Schwartz index coincides with that in [12]. We then recall the GSV-index, which is defined in [22, 9, 23]. It is defined for a vector field on a local complete intersection V in a complex manifold M and it