15 research outputs found
Differentiability of the arithmetic volume function
We introduce the positive intersection product in Arakelov geometry and prove
that the arithmetic volume function is continuously differentiable. As
applications, we compute the distribution function of the asymptotic measure of
a Hermitian line bundle and several other arithmetic invariants
Algebraic dynamical systems and Dirichlet's unit theorem on arithmetic varieties
In this paper, we study obstructions to the Dirichlet property by two
approaches: density of non-positive points and functionals on adelic
R-divisors. Applied to the algebraic dynamical systems, these results provide
examples of nef adelic arithmetic R-Cartier divisor which does not have the
Dirichlet property. We hope the obstructions obtained in the article will give
ways toward criteria of the Dirichlet property.Comment: 36 page
On the concavity of the arithmetic volumes
In this note, we study the differentiability of the arithmetic volumes along
arithmetic R-divisors, and give some equality conditions for the
Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big
arithmetic R-divisors.Comment: 35 page
Differentiability of the -volume function over an adelic curve
In this article, we show a differentiability property for the -volume
function on the ample cone of adelic line bundles over an adelic curve. This
result is deduced from a non-Archimedean counterpart of a diffrentiability
result of Witt Nystr\"om. As an application, we give a logarithmic
equidistribution result over adelic curves.Comment: 43 pages, comments are welcome! Second version, the differentiability
property is extended to the non relative case (Theorem 1.2) and allows to
give the logarithmic equidistribution result (Theorem 1.3
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
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