Simplicial Chern-Weil theory for coherent analytic sheaves, Timothy Hosgood

Abstract: The aim of this thesis is to review and improve upon an unpublished thesis by Green, whose goal was to construct Chern classes of coherent analytic sheaves in de Rham cohomology that respect the Hodge filtration. The second part of this thesis is dedicated to the construction of a categorical enrichment of the bounded derived category of complexes of coherent sheaves on an arbitrary complex manifold: the objects are ‘simplicial’ vector bundles endowed with a certain type of simplicial connection. This construction uses the theory of twisting cochains, developed in this setting by O’Brian, Toledo, and Tong. The first part is dedicated to defining a categorical lift of the Chern character in de Rham cohomology that respects the Hodge filtration, and for this we use the categorical model mentioned above. This construction can be undertaken by adapting classical Chern-Weil theory to the simplicial setting, using Dupont’s theory of fibre integration.
Link to papers: arxiv.org/abs/..., arxiv.org/abs/...
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