7,650 research outputs found
Non-archimedean canonical measures on abelian varieties
For a closed d-dimensional subvariety X of an abelian variety A and a
canonically metrized line bundle L on A, Chambert-Loir has introduced measures
on the Berkovich analytic space associated to A with
respect to the discrete valuation of the ground field. In this paper, we give
an explicit description of these canonical measures in terms of convex
geometry. We use a generalization of the tropicalization related to the Raynaud
extension of A and Mumford's construction. The results have applications to the
equidistribution of small points.Comment: Thorough revision according to the comments of the referee. To appear
in Compositi
Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros)
Chambert-Loir and Ducros have recently introduced real differential forms and
currents on Berkovich spaces. In these notes, we survey this new theory and we
will compare it with tropical algebraic geometry.Comment: 25 pages, notes for my survey talk given at the Simons Symposium in
St. John from 1.4-5.4.2013. In the second version, a sign error in the
definition of the integral is corrected and the exposition in section 7 is
slightly change
The Bogomolov conjecture for totally degenerate abelian varieties
We prove the Bogomolov conjecture for an abelian variety A over a function
field which is totally degenerate at a place v. We adapt Zhang's proof of the
number field case replacing the complex analytic tools by tropical analytic
geometry. A key step is the tropical equidistribution theorem for A at the
totally degenerate place. As an application, we obtain finiteness of torsion
points with coordinates in the maximal unramified algebraic extension over v.Comment: 21 pages; submitted. Minor errors corrected, applications in Section
6 adde
Local heights of toric varieties over non-archimedean fields
We generalize results about local heights previously proved in the case of
discrete absolute values to arbitrary non-archimedean absolute values of rank
1. First, this is done for the induction formula of Chambert-Loir and
Thuillier. Then we prove the formula of Burgos--Philippon--Sombra for the toric
local height of a proper normal toric variety in this more general setting. We
apply the corresponding formula for Moriwaki's global heights over a finitely
generated field to a fibration which is generically toric. We illustrate the
last result in a natural example where non-discrete non-archimedean absolute
values really matter.Comment: 67 pages. v2: Assumption in Theorem 2.5.8 corrected to support
function; other minor change
Moments of meson spectral functions in vacuum and nuclear matter
Moments of the meson spectral function in vacuum and in nuclear matter
are analyzed, combining a model based on chiral SU(3) effective field theory
(with kaonic degrees of freedom) and finite-energy QCD sum rules. For the
vacuum we show that the spectral density is strongly constrained by a recent
accurate measurement of the cross section. In nuclear
matter the spectrum is modified by interactions of the decay kaons with
the surrounding nuclear medium, leading to a significant broadening and an
asymmetric deformation of the meson peak. We demonstrate that both in
vacuum and nuclear matter, the first two moments of the spectral function are
compatible with finite-energy QCD sum rules. A brief discussion of the
next-higher spectral moment involving strange four-quark condensates is also
presented.Comment: 7 pages, 5 figures; published versio
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