Let X/K be an absolutely simple abelian variety over a number field; we study whether the reductions Xp tend to be simple, too. We show that if End(X) is a definite quaternion algebra, then the reduction Xp is geometrically isogenous to the self-product of an absolutely simple abelian variety for p in a set of positive density, while if X is of Mumford type, then Xp is simple for almost all p. ...

Given a family of Abelian varieties over a positive-dimensional base, we prove that for a sufficiently general curve in the base, every rational section of the family over the curve is contained in a unique rational section over the entire base. 1. Main results sec-intro The starting point for this article is the following theorem. Theorem 1.1. [GHMS05, Theorem 6.2] Let B be a smooth, quasi-pro...

A Kuga fiber variety is a family of abelian varieties parametrized by a locally symmetric space and is constructed by using an equivariant holomorphic map of Hermitian symmetric domains. We construct a complex torus bundle T over a Kuga fiber variety Y parametrized by X and express its cohomology H∗(T ,C) in terms of the cohomology of Y as well as in terms of the cohomology of the locally symme...

Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We start with an introduction to Hodge structures and we give a detailed account of the construction of Kuga-Satake varieties. The Hodge conjecture is discussed in section 2. An excellent survey of the Hodge conjecture for abelian varieties is [...

We show that if X is a toric scheme over a regular ring containing a field of finite characteristic then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic 0. The affine case of our result was conjectured by Gubeladze.

The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V. I. Arnold and V. A. Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.

This is an introductory exposition to background material useful to appreciate various formulations of the Mordell–Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebro-geometric models (rather than model-theoretic ones) with applications (...

We study, as hypersurfaces in toric varieties, elliptic Calabi–Yau fourfolds for F-theory compactifications dual to E8 × E8 heterotic strings compactified to four dimensions on elliptic Calabi–Yau threefolds with some choice of vector bundle. We describe how to read off the vector bundle data for the heterotic compactification from the toric data of the fourfold. This map allows us to construct...

Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish th...