Chern character for infinity vector bundles

TL;DR

This paper extends the Chern character to infinity vector bundles on complex manifolds, using infinity-sheafification, enabling applications to stacks, equivariant settings, and introducing new invariants.

Contribution

It defines a Chern character for infinity vector bundles via simplicial presheaves, generalizing previous work to stacks and higher homotopy invariants.

Findings

Extension of Chern character to stacks and equivariant cases

Introduction of new Chern-Simons invariants for coherent sheaves

Framework for higher homotopy group invariants

Abstract

Coherent sheaves on general complex manifolds do not necessarily have resolutions by finite complexes of vector bundles. However D. Toledo and Y.L.L. Tong showed that one can resolve coherent sheaves by objects analogous to chain complexes of holomorphic vector bundles, whose cocycle relations are governed by a coherent infinite system of homotopies. In the modern language such objects are obtained by the infinity-sheafification of the simplicial presheaf of chain complexes of holomorphic vector bundles. We define a Chern character as a map of simplicial presheaves, whereby the connected components of its sheafification recovers the Chern character of Toledo and Tong. As a consequence our construction extends Toledo Tong and O'Brian Toledo Tong's definition of the Chern character to the settings of stacks and in particular the equivariant setting. Even in the classical setting of complex manifolds, the induced maps on higher homotopy groups provide new Chern-Simons, and higher Chern-Simons, invariants for coherent sheaves.