Institute of Mathematics

Abstract:  We construct three simplicial presheaves on the site of ringed spaces, and in particular on that of complex manifolds. The descent objects for these simplicial presheaves yield Toledo-Tong’s twisting cochains, simplicial twisting cochains, and complexes that appear in Green’s thesis on Chern classes for coherent analytic sheaves, respectively. We thus extend the aforementioned constructions to the equivariant setting, and more generally to stacks. Although this is the first step in achieving push-forwards in $K$-theory and Riemann-Roch theorems for appropriate stacks, continuing a programme of Toledo and Tong, and O'Brian and Green, this paper is one of higher category theory in the language of simplicial sets and $dg$-categories, and thus does not involve any technicalities of complex geometry.