Related papers: Horospherical varieties with quotient singularitie…

In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have…

I discuss a special class of singularities obtained as a natural 4-dimensional generalization of the conical singularity. Such singularities (called quasiregular) are ruinous for the predictive force of general relativity, so one often…

We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp(X)$, which we conjecture to satisfy the inequality $\wp(X) \le \operatorname{dim} X - \operatorname{rank} X$ with equality holding if and only if…

We reprove and generalize the result that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a…

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open set of stable points,…

We investigate the behavior of semi-orthogonal decompositions of bounded derived categories of singular varieties under flat deformations to smooth varieties. We consider a Q-Gorenstein smoothing of a surface with a quotient singularity,…

We prove the following result: Let B be a smooth, irreducible, quasi-projective variety over the complex numbers and assume that B has a projective compactification \bar{B} such that \bar{B} - B is of codimension at least two in \bar{B}.…

We obtain some results that answer certain questions of Lorenzini on wild quotient singularities in dimension two. Using Kato's theory of log structures and log regularity, we prove that the dual graph of exceptional curves on the…

We describe the Minimal Model Program in the family of $\mathbb{Q}$-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In…

We discuss two properties of an abelian variety, namely, being a direct summand in a product of Jacobians and the weaker property of being "split". We relate the first property to the integral Hodge conjecture for curve classes on abelian…

Given an orbifold, we construct an orthogonal spectrum representing its stable global homotopy type. Orthogonal spectra now represent orbifold cohomology theories which automatically satisfy certain properties as additivity and the…

We study smooth curves on abelian surfaces, especially for genus 4, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 hyperelliptic curve on a general (1,…

Let $G$ be a finite group, $X$ be a smooth complex projective variety with a faithful $G$-action, and $Y$ be a resolution of singularities of $X/G$. Larsen and Lunts asked whether $[X/G]-[Y]$ is divisible by $[\mathbb{A}^1]$ in the…

In this article, we give a complete description of the characteristic polynomials of supersingular abelian varieties over finite fields. We list them for the dimensions upto 7.

We prove measure rigidity for the action of (maximal) horospherical subgroups on homogeneous spaces obtained by quotient by a uniform (nonuniform) arithmetic lattices over a field of positive characteristic.

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical…

This article considers some affine algebraic varieties attached to finite trees and closely related to cluster algebras. Their definition involves a canonical coloring of vertices of trees into three colors. These varieties are proved to be…

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many…

This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal…

Let G be a complex connected semisimple group, whose simple components have type A or D. We prove that wonderful G-varieties are classified by means of combinatorial objects called spherical systems. This is a generalization of a known…