31 research outputs found
An arithmetic Hilbert-Samuel theorem for singular hermitian line bundles and cusp forms
We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular
hermitian line bundles of finite height. In particular, the theorem applies to
the log-singular metrics of Burgos-Kramer-K\"uhn. Our theorem is thus suitable
for application to some non-compact Shimura varieties with their bundles of
cusp forms. As an application, we treat the case of Hilbert modular surfaces,
establishing an arithmetic analogue of the classical result expressing the
dimensions of spaces of cusp forms in terms of special values of Dedekind zeta
functions
Deligne pairings and families of rank one local systems on algebraic curves
For smooth families of projective algebraic curves, we extend the notion of
intersection pairing of metrized line bundles to a pairing on line bundles with
flat relative connections. In this setting, we prove the existence of a
canonical and functorial "intersection" connection on the Deligne pairing. A
relationship is found with the holomorphic extension of analytic torsion, and
in the case of trivial fibrations we show that the Deligne isomorphism is flat
with respect to the connections we construct. Finally, we give an application
to the construction of a meromorphic connection on the hyperholomorphic line
bundle over the twistor space of rank one flat connections on a Riemann
surface.Comment: 48 pp. 1 figur
BCOV invariants of Calabi--Yau manifolds and degenerations of Hodge structures
Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part
because of mirror symmetry and enumerative geometry. After
Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve
counting on a Calabi--Yau manifold is related to a conjectured invariant, only
depending on the complex structure of the mirror, and built from Ray--Singer
holomorphic analytic torsions. To this end, extending work of
Fang--Lu--Yoshikawa in dimension 3, we introduce and study the BCOV invariant
of Calabi--Yau manifolds of arbitrary dimension. To determine it, knowledge of
its behaviour at the boundary of moduli spaces is imperative. We address this
problem by proving precise asymptotics along one-parameter degenerations, in
terms of topological data and intersection theory. Central to the approach are
new results on degenerations of metrics on Hodge bundles, combined with
information on the singularities of Quillen metrics in our previous work.Comment: Minor revision. Mainly restructure of the text, minor improvements
and corrections. Added information about subdominant terms of -norm
Hermitian structures on the derived category of coherent sheaves
The main objective of the present paper is to set up the theoretical basis
and the language needed to deal with the problem of direct images of hermitian
vector bundles for projective non-necessarily smooth morphisms. To this end, we
first define hermitian structures on the objects of the bounded derived
category of coherent sheaves on a smooth complex variety. Secondly we extend
the theory of Bott-Chern classes to these hermitian structures. Finally we
introduce the category \oSm_{\ast/\CC} whose morphisms are projective
morphisms with a hermitian structure on the relative tangent complex