Related papers: Points de petite hauteur sur les varietes semi-abe…

The basic theory of semi-measures on locally compact Abelian groups is extended to prove the existence of a generalised Eberlein decomposition into such semi-measures.

We prove asymptotics for semi-integral points of bounded height on toric varieties. We verify the Manin-type conjecture of Pieropan, Smeets, Tanimoto and V\'arilly-Alvarado for smooth and certain singular toric orbifolds upon replacing the…

Given subvarieties $X, Y$ of a complex algebraic variety $S$ of complementary dimension, must they intersect? When $S$ is projective space, this is a consequence of the classical B\'ezout theorem, and an analogue for simple abelian…

The goal of this note is to study a conjectural picture on lower bounds of Seshadri constants of indecomposable polarized abelian varieties. This is inspired by some ideas of Debarre on the subject and the author's previous work on syzygies…

For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither zero nor one. We quantify this statement, following work by V. Kolyada, and obtain the…

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…

We study N\'eron models of pseudo-Abelian varieties over excellent discrete valuation rings of equal characteristic $p>0$ and generalize the notions of good reduction and semiabelian reduction to such algebraic groups. We prove that the…

The conjecture that semi-p-abelian groups is strongly semi-p-abelian is flase for p=3.And it's true for metabelian semi-p-abelian groups.

The main purpose of this paper is to prove the following theorem on the defect relations for ample divisors of abelian varieties. Main Theorem. Let $A$ be an abelian variety of complex dimension $n$ and $D$ be an ample divisor in $A$. Let…

We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving…

We associate to an analytic subvariety of a torus a tropical variety. In the first part, we generalize the results from tropical algebraic geometry to this non-archimedean analytic situation. The periodic case is applied to a totally…

In a paper of Tate and the author, we conjectured a uniform bound for the p-adic distance of torsion points on a semiabelian variety, not lying in a subvariety, to that subvariety. We survey the progress made on that conjecture and on…

Let $A$ be a simple abelian variety over a number field $k$ such that $\operatorname{End}(A)$ is noncommutative. We show that $A$ splits modulo all but finitely many primes of $k$. We prove this by considering the subalgebras of…

We prove the discreteness of algebraic points (with respect to the Neron-Tate height) on a curve of genus greater than one embedded in his jacobian. This result was conjectured by Bogomolov. We also prove the positivity of the self…

We compare general inequalities between invariants of number fields and invariants of abelian varieties over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded…

We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities on an irregular variety $X$ are related to some natural defined birational…

We discuss Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications to hyperelliptic jacobians are given. In particular, we improve some of our earlier results.

In characteristic zero, it was proven a long time ago by D. Lieberman that cohomological and numerical equivalence coincide for cycles on abelian varieties. In this paper we show this to be true also in a somewhat technical sense for…

In the present article, we define a notion of good height functions on quasi-projective varieties $V$ defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions…

We study the notion of $1$-twisted semi-homogeneous vector bundles on $\mathbb{G}_m$-gerbes over abelian varieties, and classify point objects in the twisted derived categories of abelian varieties. As an application, we classify the…