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.