Tim Hosgood: Holomorphic Deligne cohomology

Abstract: Given a complex-analytic manifold, the constant sheaf \(\mathbb{C}\) has two important structures: the integral structure, coming from \(\mathbb{Z}\); and the Hodge structure, coming from the resolution by the de Rham complex. If we take cohomology with respect to the former, then we obtain singular cohomology, which describes the topological information of our space; if we take cohomology with respect to the latter, then we obtain de Rham cohomology, which describes the geometric information of our space. Deligne cohomology is a theory which contains both of these structures at once, and gives very powerful tools for understanding the intricate interplay between topological and geometric objects.

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