Indices for 1-Forms

Abstract

When considering smooth (real) manifolds, the tangent and cotangent bundles are isomorphic and it does not make much difference to consider either vector fields or 1-forms in order to define their indices and their relations with characteristic classes. When the ambient space is a complex manifold, this is no longer the case, but there are still ways for comparing indices of vector fields and 1-forms, and to use these to study Chern classes of manifolds. To some extent this is also true for singular varieties, but there are however important differences and each of the two settings has its own advantages.

In this chapter we briefly review the various indices of 1-forms on singular varieties through the light of the indices of vector fields discussed earlier. We define in that way the Schwartz index, the radial index, the GSV index, the homological index and the local Euler obstruction, and we study some of their relations and properties.

In this short presentation we include work done by various authors, particularly by W. Ebeling and S. Gusein-Zade, as well as ourselves in [36]. In the last section we discuss briefly the “indices of collections of 1-forms” introduced by W. Ebeling and S. Gusein-Zade: just as the index of a 1-form corresponds to the “top Chern class” (of a manifold or of a singular variety, in a sense that will be made precise in later chapters), so too the indices of collections of 1-forms correspond to other Chern numbers.

Let us mention that in his book [138], J. Schürmann introduces methods to studying singular varieties via micro-local analysis, and part of what we say below can also be considered in that framework.