Extendable Cohomologies for Complex Analytic Varieties

TL;DR

This paper introduces extendable cohomology for complex singular varieties, enabling computation of Chern classes and establishing an abstract residue theorem, with applications to holomorphic foliations.

Contribution

It develops a new cohomology theory for complex singular varieties and links it to topological invariants via a Chern-Weil approach, extending classical results to singular settings.

Findings

Defined extendable cohomology for singular varieties.

Connected extendable Chern classes with topological Chern classes.

Proved an abstract residue theorem and a Camacho-Sad type index theorem.

Abstract

We introduce a cohomology, called extendable cohomology, for abstract complex singular varieties based on suitable differential forms. Beside a study of the general properties of such a cohomology, we show that, given a complex vector bundle, one can compute its topological Chern classes using the extendable Chern classes, defined via a Chern-Weil type theory. We also prove that the localizations of the extendable Chern classes represent the localizations of the respective topological Chern classes, thus obtaining an abstract residue theorem for compact singular complex analytic varieties. As an application of our theory, we prove a Camacho-Sad type index theorem for holomorphic foliations of singular complex varieties.