The Virtual Index

Abstract

The virtual index was first introduced in [111] by D. Lehmann, M. Soares and T. Suwa for holomorphic vector fields; the extension to continuous vector fields is immediate and has been done in [30, 31, 149]. If the variety has only isolated singularities, the virtual index and the GSV index coincide. The virtual index has several interesting features, as for instance that it is relatively easy to compute when the vector field we deal with is holomorphic, and also that it is defined for vector fields with singular set a compact set of arbitrary dimension.

In this chapter we introduce the virtual index in the context of singular varieties V which are local complete intersections defined by a section of a holomorphic vector bundle N over a complex manifold M (see Sect. 5.1 below). The virtual tangent bundle is then defined as (TM − N)|V , where TM denotes the holomorphic tangent bundle of M.

One can think of the virtual index as being a localization of the top dimensional Chern class of the virtual tangent bundle, called virtual class, just as the local index of Poincaré–Hopf is a localization of the top Chern class of a manifold. The virtual index is in fact a residue which is the local contribution, relatively to a vector field v, of the top virtual class.

In Sect. 2, we show that Chern–Weil theory is very well adapted to this situation, in the framework of Čech–de Rham cohomology, and Sect. 3 is devoted to the study of residues in this context. The properties of the virtual index are detailed in the last sections of the chapter, in particular we prove an integral formula for the virtual index.