Chern character for infinity vector bundles

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 modern language, such objects are obtained by the 1-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 recover the Chern character of Toledo and Tong.As a consequence, our construction extends O'Brian, Toledo and 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.19L10; 14F06, 18F20, 58J28 1. Introduction 4940 2. The simplicial presheaves IVB and 4944 3. Chern character ChW IVB !4951 4. A higher Chern character for coherent sheaves 4959 5.The induced map on classifying stacks 4971 Appendix A. A weak equivalence sSet.y ; K/ !sSet.; K/ 4979 Appendix B. Explicit description of totalization 4982 Appendix C. Totalization and fibrant objects 4986