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Abstract

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