Related papers: Differentiability of the $\chi$-volume function ov…
In the setting of Arakelov geometry over adelic curves, we introduce the $\chi$-volume function and show some general properties. This article is dedicated to talk about the continuity of $\chi$-volume function. By discussing its…
Differentiability of geometric and arithmetic volumes of Hermitian line-bundles leads to the proof of equidistribution results on projective varieties using the variational principle. In this article, we work in the setting of adelic…
By giving an estimate on the minimal slopes, we prove a Hilbert-Samuel formula for semiample and semipositive adelic line bundles. We also show the birational invariance of the arithmetic {\chi}-volume and its continuous extension on the…
In this article, we prove the boundedness of minimal slopes of adelic line bundles over function fields of characteristic 0. This can be applied to prove the equidistribution of generic and small points with respect to a big and…
Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich…
For divisors over smooth projective varieties, we show that the volume can be characterized by the duality between pseudo-effective cone of divisors and movable cone of curves. Inspired by this result, we give and study a natural…
In this paper, we show that the arithmetic volume function defined on the space of pairs of adelic R-Cartier divisors and base conditions is differentiable at a big pair, and that its derivative is given by an arithmetic restricted positive…
In this article, we generalize several fundamental results for arithmetic divisors, such as the continuity of the volume function, the generalized Hodge index theorem, Fujita's approximation theorem for arithmetic divisors and Zariski…
By some result on the study of arithemtic over trivially valued field, we find its applications to Arakelov geometry over adelic curves. We prove a partial result of the continuity of arithmetic $\chi$-volume along semiample divisors.…
In this article, we introduce the notion of global adelic space of an arithmetic variety over an adelic curve and prove an equidistribution theorem for a generic sequence of subvarieties. As an application, we prove a Bogomolov type theorem…
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…
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…
We introduce an adelic Cartier divisor over a trivially valued field and discuss the bigness of it. For bigness, we give the integral representation of the arithmetic volume and prove the existence of limit of it. Moreover, we show that the…
We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its…
In this paper, we collect some fundamental properties of the arithmetic restricted volumes (or the arithmetic multiplicities) of the adelically metrized line bundles. The arithmetic restricted volume has the concavity property and…
This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for ample divisor classes. The second…
This work is motivated by two central questions in the birational geometry of moduli spaces of curves -- Fulton's conjecture and the effective cone of $\bar M_g$. We study the algebro-geometric aspect of Teichmuller curves parameterizing…
Let $L$ be a big holomorphic line bundle on a complex projective manifold $X.$ We show how to associate a convex function on the Okounkov body of $L$ to any continuous metric $\psi$ on $L.$ We will call this the Chebyshev transform of…
Let $\mathcal X$ be a projective arithmetic variety of dimension at least $2$. If $\overline{\mathcal L}$ is an ample hermitian line bundle on $\mathcal X$, we prove that the proportion of those effective sections of $\overline{\mathcal…
We prove that `volume cone implies metric cone' in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding valid in Ricci-limit spaces.