Dustin Clausen - 3/4 Three perspectives on Deligne cohomology

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Dustin Clausen: Title: Three perspectives on Deligne cohomology Abstract: Deligne cohomology is a refined cohomology theory for complex manifolds, known in number theory for its role both in Arakelov geometry and in Beilinson&#39;s conjectures on special values of L-functions. In these lectures I will describe three interrelated perspectives on Deligne cohomology. The first represents an attempt to determine for which kinds of &quot;analytic spaces&quot; the standard definition of Deligne cohomology makes sense and has good properties. The second and third, which describe theories only conjecturally equivalent to Deligne cohomology, represent attempts to articulate that Deligne cohomology is the complex-analytic analog of motivic cohomology. In the second, which is joint with with Peter Scholze, we define (using Efimov&#39;s continuous K-theory) a version of algebraic K-theory for analytic spaces and conjecture that it identifies with the (easily defined) K-theory analog of Deligne cohomology. In the third we more directly articulate a conjectural universal property for Deligne cohomology inspired by the recent &quot;non-A^1-invariant motivic&quot; formalism of Annala-Iwasa in the algebraic setting. The first and second perspectives make use of the theory of analytic geometry developed in joint work with Peter Scholze; we will review the necessary aspects of that theory.

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