Joint with Cheyne Glass, Micah Miller, and Thomas Tradler.

In the first part of the talk, I will describe the formulae for the Hodge and de Rham characteristic classes of holomorphic bundles solely in terms of their clutching functions. To do so, I define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable isomorphisms of vector bundles with holomorphic connections that do not necessarily preserve the connections, an appropriate sequence of holomorphic forms. We then apply this map of simplicial presheaves to the Cech nerve of a cover of a complex manifold and assemble the data by passing to the totalization. In this way, we obtain a map of simplicial sets that on the vertices produces an explicit formula for the Hodge Chern character of a bundle in terms of its clutching functions. On the edges, we obtain similar invariants associated to isomorphisms between bundles. Similarly, we obtain suitable Hodge and de Rham Chern characters in the equivariant setting.

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