相关论文: Singularities of divisors of low degree on abelian…

We use the method of Adler-van Moerbeke and Vanhaecke to show that the general fibre of the hamiltonian system of Dorizzi, Grammaticos and Ramani completes to a (1, 6)-polarised abelian surface.

Let A be the moduli space of (1,p)-polarised abelian surfaces with a level structure, for p an odd prime. Let X be a desingularisation of any algebraic compactification of A. Then X is simply-connected.

We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by…

We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.

Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces having real multiplication. Similar to…

Necessary and sufficient conditions are given for a $G$-graded simple module over a unital associative algebra, graded by an abelian group $G$, to be isomorphic to a loop module of a simple module, as well as for two such loop modules to be…

We study the volumes of divisors in Calabi--Yau type varieties. We show that given a klt Calabi--Yau pair $(X,B)$ and an integral divisor $A$ on $X$, the volume of $A$ is in a fixed discrete set depending only on the dimension and…

We give a precise classification, in terms of Shimura data, of all 1-dimensional Shimura subvarieties of a moduli space of polarized abelian varieties.

We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge…

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of…

Let f be a weight two newform for Gamma_1(N) without complex multiplication. In this article we study the conductor of the absolutely simple factors B of the variety A_f over certain number fields L. The strategy we follow is to compute the…

This note discusses some examples showing that the crystalline cohomology of even very mildly singular projective varieties tends to be quite large. In particular, any singular projective variety with at worst ordinary double points has…

Let $\lambda\colon A\rightarrow A^{\vee}$ be a polarization on an abelian variety over a field $k$. If $k$ is not algebraically closed, there might not exist an ample line bundle on $A$ defined over $k$ that represents $\lambda$. To remedy…

We study semistable reduction and torsion points of abelian varieties. In particular, we give necessary and sufficient conditions for an abelian variety to have semistable reduction. We also study N\'eron models of abelian varieties with…

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 show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields,…

It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating…

The local invariants of a meromorphic Abelian differential on a Riemann surface of genus $g$ are the orders of zeros and poles, and the residues at the poles. The main result of this paper is that with few exceptions, every pattern of…

We prove that if $g\geq 2$ then the set of all Abelian differentials $(M,\omega)$ for which the vertical flow is mildly mixing is dense in every stratum of the moduli space $\mathcal{H}_g$. The proof is based on a sufficient condition for…

This is the sequel to arXiv:math/0001089. In this paper, we complete the promised description of moduli of abelian surfaces of low degree, covering the cases of degree (1,12), (1,14), (1,16), (1,18) and (1,20). In each case, we describe…