Hermitian-holomorphic Deligne cohomology, Deligne pairing for singular metrics, and hyperbolic metrics

Details Hermitian-holomorphic Deligne cohomology, Deligne pairing for singular metrics, and hyperbolic metrics Hermitian-holomorphic Deligne cohomology, Deligne pairing for singular metrics, and hyperbolic metrics

2004-08-09

arxiv.org/abs/math/0408118v2

Int. Mathematics Research Notices 17 (2005), 1015--1046

Mathematics

Algebraic Geometry

LaTeX + amsmath + mathrsfs + textcomp. 21 pages, no figures Added one clarifying lemma at the end, no changes to the main resu

Scientific paper

We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to be a functional whose extrema are the metrics of constant negative curvature.

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