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 $c_1(L|_X)^{\wedge d}$ 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 ϕ\phi meson spectral functions in vacuum and nuclear matter

Moments of the $\phi$ 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 $e^+ e^- \to K^+ K^-$ cross section. In nuclear matter the $\phi$ 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 $\phi$ 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