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Coherent sheaves on general complex manifolds do not necessarily haveresolutions 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 chaincomplexes of holomorphic vector bundles, whose cocycle relations are governedby a coherent infinite system of homotopies. In the modern language suchobjects are obtained by the infinity-sheafification of the simplicial presheafof chain complexes of holomorphic vector bundles. We define a Chern characteras a map of simplicial presheaves, whereby the connected components of itssheafification recovers the Chern character of Toledo and Tong. As aconsequence our construction extends Toledo Tong and O'Brian Toledo Tong'sdefinition of the Chern character to the settings of stacks and in particularthe equivariant setting. Even in the classical setting of complex manifolds,the induced maps on higher homotopy groups provide new Chern-Simons, and higherChern-Simons, invariants for coherent sheaves.

Chern character for infinity vector bundles