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

We define a map of simplicial presheaves, the Chern character, that assignsto every sequence of composable non connection preserving isomorphisms ofvector bundles with holomorphic connections an appropriate sequence ofholomorphic forms. We apply this Chern character map to the Cech nerve of agood cover of a complex manifold and assemble the data by passing to thetotalization to obtain a map of simplicial sets. In simplicial degree 0, thismap gives a formula for the Chern character of a bundle in terms of theclutching functions. In simplicial degree 1, this map gives a formula for theChern character of bundle maps. In each simplicial degree beyond 1, theseinvariants, defined in terms of the transition functions, govern thecompatibilities between the invariants assigned in previous simplicial degrees.In addition to this, we also apply this Chern character to complex Liegroupoids to obtain invariants of bundles on them in terms of the simplicialdata. For group actions, these invariants land in suitable complexescalculating various Hodge equivariant cohomologies. In contrast, the de RhamChern character formula involves additional terms and will appear in a sequelpaper. In a sense, these constructions build on a point of view of"characteristic classes in terms of transition functions" advocated by RaoulBott, which has been addressed over the years in various forms and degrees,concerning the existence of formulae for the Hodge and de Rham characteristicclasses of bundles solely in terms of their clutching functions.