Simplicial Chern-Weil theory for coherent analytic sheaves, part II

TL;DR

This paper extends simplicial Chern-Weil theory to coherent analytic sheaves, demonstrating that Green's barycentric simplicial connection is admissible and enabling the construction of characteristic classes.

Contribution

It proves the admissibility of Green's barycentric simplicial connection and applies Chern-Weil theory to define Chern classes for coherent analytic sheaves.

Findings

Green's barycentric simplicial connection is admissible.

The simplicial construction matches explicit ech representatives of Atiyah classes.

Chern classes of coherent analytic sheaves are well-defined and unique in compact cases.

Abstract

In the previous part of this diptych, we defined the notion of an admissible simplicial connection, as well as explaining how H.I. Green constructed a resolution of coherent analytic sheaves by locally free sheaves on the \v{C}ech nerve. This paper seeks to apply these abstract formalisms, by showing that Green's barycentric simplicial connection is indeed admissible, and that this condition is exactly what we need in order to be able to apply Chern-Weil theory and construct characteristic classes. We show that, in the case of (global) vector bundles, the simplicial construction agrees with what one might construct manually: the explicit \v{C}ech representatives of the exponential Atiyah classes of a vector bundle agree. Finally, we summarise how all the preceding theory fits together to allow us to define Chern classes of coherent analytic sheaves, as well as showing uniqueness in the compact case.