Related papers: Slope boundedness and Equidistribution theorem
We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from…
Given a finite volume hyperbolic surface, a fundamental polygon and an oriented closed geodesic, we associate a partial covering of the surface. We prove that given a sequence of collections of oriented closed geodesics equidistributing in…
A uniform bound of intersection multiplicities of curves and divisors on abelian varieties is proved by algebraic geometric methods. It extends and improves a result obtained by A. Buium with a different method based on Kolchin's…
This is the second paper of a series. It extends the results of the first paper from number fields to finitely generated fields, based on the recent theory of adelic line bundles of the same authors. We prove an arithmetic Hodge index…
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…
In this paper, the equidistribution theorem of Szpiro-Ullmo-Zhang about sequences of small points in an abelian variety is extended to the case of sequences of higher dimensional subvarieties. A quantitative version of this result is also…
We prove an analog of the Szemer\'edi-Trotter theorem in the plane for definable curves and points in any o-minimal structure over an arbitrary real closed field $\mathrm{R}$. One new ingredient in the proof is an extension of the well…
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…
A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding…
We prove uniform boundedness statements for semistable pure sheaves on projective manifolds. For example, we prove that the set of isomorphism classes of pure sheaves of dimension 2 that are slope semistable with respect to ample classes…
We prove an equidistribution result for small subvarieties of an abelian variety which generalizes the Szpiro-Ullmo-Zhang theorem on equidistribution of small points.
A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which…
Given several sequences of Hermitian holomorphic line bundles $\{(L_{kp}, h_{kp})\}_{p=1}^{\infty}$, we establish the distribution of common zeros of random holomorphic sections of $L_{kp}$ with respect to singular measures. We also study…
We show that C-minimal fields (i.e., C-minimal expansions of ACVF) have the exchange property, answering a question of Haskell and Macpherson. Additionally, we strengthen some theorems of Cubides Kovacsics and Delon on C-minimal fields.…
We generalize the Demailly approximation theorem from complex geometry to Arakelov geometry.As an application, let $X/\mathbb{Q}$ be an integral projective variety and $\overline N$ be an adelic line bundle on $X$, we prove that…
A necessary and sufficient condition is given for semi-ampleness of a numerically effective (nef) and big line bundle in positive characteristic. One application is to the geometry of the universal stable curve over M_g, specifically, the…
This paper explores the cohomological consequences of the existence of moduli spaces for flat bundles with bounded rank and irregularity at infinity and gives unconditional proofs. Namely, we prove the existence of a universal bound for the…
We introduce the notion of $\epsilon$-irreducibility for arithmetic cycles meaning that the degree of its analytic part is small compared to the degree of its irreducible classical part. We will show that for every $\epsilon>0$ any…
We prove that for a non-isotrivial abelian scheme over a smooth curve, the genus of a generic sequence of multi-sections with small heights tends to infinity. As an application, we give a new proof of the uniform boundedness of…
We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for sufficiently large smooth projective families $f : \mathscr{X} \to \mathscr{S}$ defined over the ring of $N$-integers…