Related papers: Kuga-Satake Abelian Varieties in Positive Characte…
We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also…
Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An…
We show that the (compactified) moduli space of abelian surfaces with a polarisation of type $(1,p^2)$ is of general type for $p \ge 11$, improving a result of O'Grady.
We introduce a stratification on the space of symplectic flags on the de Rham bundle of the universal principally polarized abelian variety in positive characteristic and study its geometric properties like irreducibility of the strata and…
Perfect quadratic forms give a toroidal compactification of the moduli space of principally polarized abelian g-folds that is Q-factorial and whose ample classes are characterized, over any base. In characteristic zero it has canonical…
By means of analytic methods the quasi-projectivity of the moduli space of algebraically polarized varieties with a not necessarily reduced complex structure is proven including the case of non-uniruled polarized varieties.
This is a survey paper on moduli spaces that have a natural structure of a (possibly incomplete) locally symmetric variety. We outline the Baily-Borel compactification for such varieties and compare it with the compactifications furnished…
We review the results on the cycle classes of the strata defined by the height and the Artin invariant on the moduli of K3 surfaces in positive characteristic obtained in joint work with Katsura and Ekedahl. In addition we prove a new…
Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3…
In this note we prove that a smooth projective variety (defined over a field $k$) of non-negative Kodaira dimension that has a $k$-rational point and a polarized self map must be a finite free quotient of an abelian variety.
We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in…
In this note, we propose the modular height of an abelian variety defined over a field of finite type over Q. Moreover, we prove its finiteness property.
The present article is concerned with mirror symmetry for generalized K3 surfaces, with particular emphasis on complex and K\"ahler rigid structures. Inspired by the works of Dolgachev, Aspinwall-Morrison and Huybrechts, we introduce a…
It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one…
We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to…
A projective hyperk\"ahler manifold of Kummer-type is said to be twisted modular if it is birational to the Albanese fiber of a moduli space of twisted sheaves on an abelian surface. We prove that, with the exception of certain cases of…
In their paper Livn\'e and Yui (math.AG/0304497) discuss several examples of non-rigid Calabi-Yau varieties which admit semi-stable K3-fibrations with 6 singular fibres over a base which is a rational modular curve. They also establish the…
We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalises Kodaira's classification of singular elliptic fibres and thus call them extended ADE curves. On such a curve $C$,…
We show that there are uncountably many algebraic extensions of $\mathbb{Q}$ containing at most finitely many moduli of CM simple principally polarized abelian varieties of any fixed dimension $g\geqslant1$, generalizing a result of…
Let k be a field of characteristic different from 2. There can be an obstruction for an indecomposable principally polarized abelian threefold (A,a) over k to be a Jacobian over k. It can be computed in terms of the rationality of the…