AN ADJUNCTION FORMULA FOR LOCAL COMPLETE INTERSECTIONS
AN ADJUNCTION FORMULA FOR LOCAL COMPLETE INTERSECTIONS
复制标题
局部完全交点的并算公式
DOI：
10.1142/s0129167x98000324
复制
发表时间：
1998
影响因子：
0.6
通讯作者：
T. Suwa
中科院分区：
文献类型：
--
作者：
J. Seade;T. Suwa
本文研究了奇异簇上向量场的各种指标，并作为应用，证明了对于具有孤立奇点的紧“强”局部完全交V，用V的虚切丛的顶Chern类和奇点的Milnor数表示Euler-Poincare特征线x（^）0 I ^的公式（定理2.4）.对于在奇异簇V上的向量场，我们考虑了在V的奇性处的“Schwartz指标”、“GSV指标”和“虚指标”，当V的奇性在V的正则部分时，这些指标都归结为通常的Poincare-Hopf指标，因此我们比较了它们在V的奇性部分时的情况。H. Schwartz定义了奇异簇V上的“径向”向量场的指数，见[21，4]。当V的奇点是孤立的时，就像本文中的那样，这个定义可以很容易地扩展到非径向的向量场。我们在SEC这样做。1，我们称相应的指数为向量场的Schwartz指数。我们证明了，对于紧致簇V上具有孤立奇点的整体向量场，Schwartz指标之和给出x（V）（定理1.2）。在[12]中，定义了奇异簇上的分层向量场的局部指标，推广了Schwartz关于径向向量场的定义。假设我们对Schwartz指数的定义与[12]中的一致。然后我们回顾GSV索引，它在[22，9，23]中定义。它定义为复流形M中局部完全交V上的向量场，
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 virtual 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 "Schwartz index", the "GSV-index" and the "virtual index" 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 "radial" 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