Related papers: Deformations of $\mathbb{A}^1$-cylindrical varieti…

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a…

We know that semi-regular sub-varieties satisfy the variational Hodge conjecture i.e., given a family of smooth projective varieties $\pi:\mathcal{X} \to B$, a special fiber $\mathcal{X}_o$ and a semi-regular subvariety $Z \subset…

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ for (affine) algebraic varieties $X$ and $X'$ implies that $X\cong X'$. In this paper we provide a…

We consider some algebraic aspects of the dynamics of an automorphism on a family of polarized abelian varieties parameterized by the complex unit disk. When the action on the cohomology of the generic fiber has no cyclotomic factor, we…

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…

In this paper we explore conditions for a curve in a smooth projective surface to have a free product of cyclic groups as the fundamental group of its complement. It is known that if the surface is $\mathbb P^2$, then such curves must be of…

A cellular algebra is called cyclic cellular if all cell modules are cyclic. Most important examples of cellular algebras appearing in representation theory are in fact cyclic cellular. We prove that if $A$ is a cyclic cellular algebra,…

Let $G$ be an algebraic group and let $X$ be a smooth $G$-variety with two orbits: an open orbit and a a closed orbit of codimension $1$. We give an algebraic description of the category of $G$-equivariant vector bundles on $X$ under a mild…

For a real algebraic K3 surface $X(R)$, we give all possible values of the dimension $h^1_{alg}(X(R)$ of the group $\H^1_{alg}(X(R),Z/2)$ of algebraic cycles of $X(R)$. In particular, we prove that if $X(R)$ is not an M-surface, $X(R)$ can…

We prove that the universal cover of a normal complex algebraic variety admitting a faithful complex representation of its fundamental group is an analytic Zariski open subset of a holomorphically convex complex space. This is a non-proper…

The geometric condition defining a spherical variety for a reductive algebraic group was generalized in [AG21], with applications to representation theory. We twist by a character to generalize this definition, and show its equivalence to a…

Classically, regular homomorphisms have been defined as a replacement for Abel--Jacobi maps for smooth varieties over an algebraically closed field. In this work, we interpret regular homomorphisms as morphisms from the functor of families…

This paper stems from the observation (arising from work of T. Delzant) that "most" K\"ahler groups virtually algebraically fiber, i.e. admit a finite index subgroup that maps onto $\Bbb{Z}$ with finitely generated kernel. For the remaining…

As in Zariski's Uniformization Theorem we show that a valuation ring $V$ of characteristic $p>0$ of dimension one is a filtered direct limit of smooth ${\bf F}_p$-algebras under some conditions of transcendence degree. Under mild…

Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial…

In this paper we study a special class of fibrations on Delsarte surfaces. We call these fibrations Delsarte fibrations. We show that after a specific cyclic base change the fibration is the pull back of a fibration with three singular…

Let $\pi\colon \mathcal{X}\to B$ be a family whose general fiber $X_b$ gives a $(d_1,...,d_a)$ polarisation of a general Abelian variety where $1\leq d_i\leq 2$, $i=1,...,a$ and $a\geq 4$. We show that the fibers are in the same birational…

We prove thst the deformation complex of a d-algebra (shifted by 1-d) carries a natural structure of (d+1)-algebra. This is a purely algebraic version of a similkar theorem of Kontsevich.

An algebraic variety X is embedded to the order k via a line bundle L if the global sections of L generate all (simultaneous) jets of order k on X or if they separate all zero-dimensional subschemes of length at most k+1. Even though we…

We explicitly describe the Albanese morphism of a hyperelliptic variety, i.e., the quotient $X$ of an abelian variety $A$ by a finite group $G$ acting freely and not only by translations, by giving a description of the Albanese variety and…