On log canonical inversion of adjunction

We prove a result on the inversion of adjunction for log canonical pairs that generalizes Kawakita's result to log canonical centers of arbitrary codimension.Comment: To appear in the PEMS, volume in honour of V. Shokuro

Singularities of pluri-theta divisors in Char p>0p>0

We show that if $(X,\Theta)$ is a PPAV over an algebraically closed field of characteristic $p>0$ and $D\in |m\Theta|$, then $(X,\frac 1 m D)$ is a limit of strongly $F$-regular pairs and in particular ${\rm mult}_x(D)\leq m\cdot \dim X$ for any $x\in X$

Fourier transforms, generic vanishing theorems and polarizations of abelian varieties

The purpose of this paper is to give two applications of Fourier transforms and generic vanishing theorems: - we give a cohomological characterization of principal polarizations - we prove that if $X$ an abelian variety and $\Theta$ a polarization of type $(1,...,1,2)$, then a general pair $(X,\Theta )$ is log canonicalComment: 10 page

A derived category approach to generic vanishing

We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field $k$. When k=\CC this implies a conjecture of Green and Lazarsfeld.Comment: 11 pages, comments welcom

Singularities of divisors of low degree on abelian varieties

Building on previous work of Kollar, Ein, Lazarsfeld, and Hacon, we show that ample divisors of low degree on an abelian variety have mild singularities in case the abelian variety is simple or the degree of the polarization is two.Comment: Minor modifications, improved exposition. 13 page

Deformations of the trivial line bundle and vanishing theorems

This paper reproves a general form of the Green-Lazarsfeld 'generic vanishing' theorem and more recent strengthenings, as well as giving some new applications.Comment: Latex2e file, 33 page

On infinite dimensional grassmannians and their quantum deformations

An algebraic approach is developed to define and study infinite dimensional grassmannians. Using this approach a quantum deformation is obtained for both the ind-variety union of all finite dimensional grassmannians, and the Sato grassmannian introduced by Sato. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite dimensional version of the first theorem of invariant theory is discussed for both the infinite dimensional special linear group and its quantization

On the rationality of Kawamata log terminal singularities in positive characteristic

We show that there exists a natural number $p_0$ such that any three-dimensional Kawamata log terminal singularity defined over an algebraically closed field of characteristic $p>p_0$ is rational and in particular Cohen-Macaulay.Comment: An example, by Takehiko Yasuda, of a non-Cohen-Macaulay quotient klt singularity for any p>2 has been added. Some small changes are introduced. Please note, that being Cohen-Macaulay is now a part of the definition of rational singularitie

On the characterization of abelian varieties in characteristic p>0p>0

This article is superseded by 1703.06631. We keep this version here since some of the arguments for the special cases treated here are different than those of 1703.06631.Comment: This article is superseded by 1703.0663

On the Adjunction Formula for 33-folds in characteristic p>5p>5

In this article we prove a relative Kawamata-Viehweg vanishing-type theorem for PLT $3$-folds in characteristic $p>5$. We use this to prove the normality of minimal log canonical centers and the adjunction formula for codimension $2$ subvarieties on $\mathbb{Q}$-factorial $3$-folds in characteristic $p>5$.Comment: 19 Pages. Comments are welcome. To appear in the Mathematische Zeitschrif