MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

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Funded by The Danish National Research Foundation

Geometric Mathematical Physics Seminar

Abstract
The tame symbol associated to a pair of invertible holomorphic functions can be computed as a cup product in Deligne cohomology. This construction produces a line bundle equipped with an analytic connection. "Higher" versions were considered by Brylinski and McLaughlin and related to gerbes (or 2-gerbes) equipped with an appropriate notion of connective structure. These constructions can be enhanced to take into account the datum of a hermitian structure---the resulting complexes (resp. cohomology) are termed Hermitian-holomorphic Deligne (HHD) complexes (resp. cohomology). As a simple example, the group of isomorphism classes of holomorphic line bundles with hermitian metric is expressed as a degree 2 (hyper)cohomology group with values in one of the HHD complexes. After reviewing the main definitions, we show how these objects naturally appear in uniformization problems concerning constant negative curvature metrics on compact Riemann surfaces. One such metric subordinated to a given conformal structure satisfies an extremum condition. We present an algebraic construction for the appropriate functional as the square of the metrized holomorphic tangent bundle in an HHD group. For a pair of line bundles on a Riemann surface, this construction recovers the algebraic approach to the determinant of cohomology.

Contact person:Jørgen Ellegaard Andersen.