The Case of Holomorphic Vector Fields

Abstract

We have seen that for vector fields, there are indices such as the Poincaré–Hopf index and the virtual index, that arise from localizations of certain Chern classes. If the vector field is holomorphic, the localization theory becomes richer because of the Bott vanishing theorem, and this produces further interesting residues. This theory can be developed for general singular foliations on certain singular varieties. We consider here the case of holomorphic vector fields and the slightly more general case of one dimensional singular foliations. We refer to [156] for a systematical treatment of the general case.

Here we have three types of residues:

• (1) Baum–Bott residues and generalizations to singular varieties,

• (2) Camacho–Sad index and various generalizations,

• (3) Variations and generalizations.

In all the above cases the residues arise from a Bott type vanishing theorem, which in turn comes from an action of the vector field or the foliation on some vector bundle or virtual bundle. The residues of type (1) were first introduced by R. Baum and P. Bott in [13,14]. In general these arise from the action of the foliation on the normal sheaf of the foliation. The Camacho–Sad index (2) was introduced in [42] and was effectively used to prove the existence of a separatrix at a singular point of a holomorphic vector field on the complex plane. Nowadays there are many generalizations of this index, see Remark 6.3.3 below. These residues arise from the action of the foliation on the normal bundle of an invariant subvariety. The residues of type (3) were first introduced by B. Khanedani and T. Suwa in [93] and generalized in [113]; see also the related articles [39] and [40] by M. Brunella. These type of residues arise from the action of the foliation on the ambient tangent bundle. These three types of residues are listed above in historical order, but they are explained below in the reversed order, for logical reasons.

In each case, the residue at an isolated singularity can be expressed in terms of a Grothendieck residue on singular variety.