469 research outputs found
Semipurity of tempered Deligne cohomology
In this paper we define the formal and tempered Deligne cohomology groups,
that are obtained by applying the Deligne complex functor to the complexes of
formal differential forms and tempered currents respectively. We then prove the
existence of a duality between them, a vanishing theorem for the former and a
semipurity property for the latter. The motivation of these results comes from
the study of covariant arithmetic Chow groups. The semi-purity property of
tempered Deligne cohomology implies, in particular, that several definitions of
covariant arithmetic Chow groups agree for projective arithmetic varieties
Successive minima of toric height functions
Given a toric metrized R-divisor on a toric variety over a global field, we
give a formula for the essential minimum of the associated height function.
Under suitable positivity conditions, we also give formulae for all the
successive minima. We apply these results to the study, in the toric setting,
of the relation between the successive minima and other arithmetic invariants
like the height and the arithmetic volume. We also apply our formulae to
compute the successive minima for several families of examples, including
weighted projective spaces, toric bundles and translates of subtori.Comment: To appear in Annales de l'Institut Fourier (Grenoble), 40 pages, 5
figure
Arithmetic geometry of toric varieties. Metrics, measures and heights
We show that the height of a toric variety with respect to a toric metrized
line bundle can be expressed as the integral over a polytope of a certain
adelic family of concave functions. To state and prove this result, we study
the Arakelov geometry of toric varieties. In particular, we consider models
over a discrete valuation ring, metrized line bundles, and their associated
measures and heights. We show that these notions can be translated in terms of
convex analysis, and are closely related to objects like polyhedral complexes,
concave functions, real Monge-Amp\`ere measures, and Legendre-Fenchel duality.
We also present a closed formula for the integral over a polytope of a function
of one variable composed with a linear form. This allows us to compute the
height of toric varieties with respect to some interesting metrics arising from
polytopes. We also compute the height of toric projective curves with respect
to the Fubini-Study metric, and of some toric bundles.Comment: Revised version, 230 pages, 3 figure
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