Global Chern currents of coherent sheaves and Baum Bott currents

TL;DR

This paper extends the theory of characteristic class currents for coherent sheaves and holomorphic foliations, allowing for their representation by currents supported on relevant geometric subsets without requiring global resolutions.

Contribution

It introduces a method to construct characteristic class representatives using the Chern-Green approach, applicable on arbitrary complex manifolds, and establishes a new transgression formula for these currents.

Findings

Currents can be supported on the sheaf support or foliation singular set.

The approach works without global resolutions of sheaves.

A new transgression formula relates different current representatives.

Abstract

We provide global extensions of previous results about representations of characteristic classes of coherent analytic sheaves and of Baum-Bott residues of holomorphic foliations. We show in the first case that they can be represented by currents with support on the support of the given coherent analytic sheaf, and in the second case, by currents with support on the singular set of the foliation. In previous works, we have constructed such representatives provided global resolutions of the appropriate sheaves existed. In this article, we show that the definition of Chern classes of Green and the associated techniques, which work on arbitrary complex manifolds without any assumption on the existence of global resolutions, may be combined with our previous constructions to yield the desired representatives. We also prove a transgression formula for such representatives, which is new even in the case when global resolutions exist. More precisely, the representatives depend on local resolutions of the sheaf, and on choices of metrics and connections on these bundles, i.e., the currents for two different choices differ by a current of the form $dN$, where $N$ is an explicit current, which in the first case above has support on the support of the given coherent analytic sheaf, and in the second case above has support on the singular set of the foliation.