 
Summary: Hermitianholomorphic Deligne cohomology, Deligne pairing for
singular metrics, and hyperbolic metrics
Ettore Aldrovandi
Department of Mathematics
Florida State University
Tallahassee, FL 323064510, USA
aldrovandi@math.fsu.edu
Dedicated to µT
Abstract
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 cupproduct 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.
Contents
1 Introduction 2
1.1 Preliminaries and statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
