The Hodge Chern character of holomorphic connections as a map of simplicial presheaves

We define a map of simplicial presheaves, the Chern character, that assigns to every sequence composable non connection preserving isomorphisms vector bundles with holomorphic connections an appropriate forms. apply this character Cech nerve good cover complex manifold and assemble data by passing totalization obtain sets. In degree 0, gives formula for bundle in terms clutching functions. 1, maps. each beyond these invariants, defined transition functions, govern compatibilities between invariants assigned previous degrees. addition this, we also Lie groupoids on them data. For group actions, land suitable complexes calculating various Hodge equivariant cohomologies. contrast, de Rham involves additional will appear sequel paper. sense, constructions build point view "characteristic classes functions" advocated Raoul Bott, which has been addressed over years forms degrees, concerning existence formulae characteristic solely their