Related papers: Albanese and Picard 1-motives

Let $X$ be a projective variety, homogeneous under a linear algebraic group. We show that the diagonal of $X$ belongs to a unique irreducible component $H_X$ of the Hilbert scheme of $X\times X$. Moreover, $H_X$ is isomorphic to the…

We show that the statement analogous to the Mumford-Tate conjecture for abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image…

Let $1 \to N \to G \to H \to 1$ be an abelian extension. The purpose of this paper is to study the problem of extending automorphisms of $N$ and lifting automorphisms of $H$ to certain automorphisms of $G$.

We present a divisibility relation for the dimensions and Hilbert series of certain classes of Nichols algebras of non-abelian group type, which generalizes Nichols algebras over Coxeter groups with constant cocycle -1. For this we…

Let $\mathcal{X}_{\Gamma}G:=\mathrm{Hom}(\Gamma,G)/\!/G$ be the $G$-character variety of $\Gamma$, where $G$ is a complex reductive group and $\Gamma$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and…

We prove several duality theorems for the Galois and etale cohomology of 1-motives defined over local and global fields and establish a 12-term Poitou-Tate type exact sequence. The results give a common generalisation and sharpening of…

We consider algebraic groups GL_1(A), SL_1(A), where A is a division algebra of prime degree over a field F, and associated motives in the category of motivic complexes DM(F). Following an idea of Suslin, we relate motives of these groups…

We define a regulator map from the weight n polylogarithmic motivic complex to the weight n Deligne complex of an algebraic variety X. The regulator map is constructed explicitly via the classical polylogarithms with some funny combinations…

In this paper, we propose a definition of Neron models of arbitrary Deligne 1-motives over Dedekind schemes, extending Neron models of semi-abelian varieties. The key property of our Neron models is that they satisfy a generalization of…

We realize several combinatorial Hopf algebras based on set compositions, plane trees and segmented compositions in terms of noncommutative polynomials in infinitely many variables. For each of them, we describe a trialgebra structure, an…

Let $X$ be a homogeneous space of a connected linear algebraic group $G$ defined over the field of complex numbers $\mathbb C$. Let $x\in X({\mathbb C})$ be a point. We denote by $H$ the stabilizer of $x$ in $G$. When $H$ is connected, we…

Explicit generators are found for the group of automorphisms of the algebra of one-sided inverses of a polynomial algebra in $n$ variables. An analogue of the polynomial Jacobian homomorphism is found.

Through the following, we establish the conditions which allow us to express recursive sequences of real numbers, enumerated through the recurrence relation a_{n+1} = Aa_n + Ba_{n-1}, by means of algebraic equations in two variables of…

An algebraic variety $X$ is called a homogeneous space if there exists a transitive regular action of an algebraic group on $X$. We prove inequalities between the dimension of a homogeneous space of a linear algebraic group and its Picard…

Let K be an algebraically closed field of characteristic zero. We say that a polynomial automorphism f : K^n -> K^n is special if the Jacobian of f is equal to 1. We show that every (n - 1)-dimensional component H of the set Fix(f) of fixed…

Let X be a projective variety, possibly singular. A generalized Albanese variety of X was constructed by Esnault, Srinivas and Viehweg over algebraically closed base field as a universal regular quotient of the relative Chow group of…

Considering a (co)homology theory $\mathbb{T}$ on a base category $\mathcal{C}$ as a fragment of a first-order logical theory we here construct an abelian category $\mathcal{A}[\mathbb{T}]$ which is universal with respect to models of…

A characterization of the finite-dimensional Leibniz algebras with an abelian subalgebra of codimension two over a field $\mathbb{F}$ of characteristic $p\neq2$ is given. In short, a finite-dimensional Leibniz algebra of dimension $n$ with…

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…

We show that if X is a nonsingular projective variety of general type over an algebraically closed field k of positive characteristic and X has maximal Albanese dimension and the Albanese map is separable, then |4K_X| induces a birational…