The Hodge and de Rham Chern Characters of holomorphic connections
Plática dada por Mahmoud Zeinalian (Lehman College, City University of New York) en el Centro de Colaboración Samuel Gitler el lunes 18 de febrero del 2019.
Video proporcionado por Ciencias TV
We address an old question of Raoul Bott concerning the existence of combinatorial formulae for the Hodge and de Rham characteristic classes of bundles solely in terms of their clutching functions. To do so, 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 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 that produces explicit formulae for the Hodge Chern character of a bundle in terms of its clutching functions. We will discuss the de Rham theory formulae which in a sense are more complete than the Hodge theory.
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