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

Timothy Hosgood

Published 2020-03-22Version 1

In "Chern classes for coherent sheaves", H.I. Green constructs Chern classes in de Rham cohomology of coherent analytic sheaves, but in a way that is directly amenable to being abstractified to give us a theory of simplicial connections, as well as a simplicial version of Chern-Weil theory. We construct here a formal $(\infty,1)$-categorical framework into which we can place Green's work, as well as a more general idea as to what exactly a simplicial connection should be. The result will be the ability to work with generalised invariant polynomials (which will be introduced in the sequel to this paper) evaluated at the curvature of so-called admissible simplicial connections to get explicit \v{C}ech representatives in de Rham cohomology of characteristic classes of coherent analytic sheaves.