Chevalley’s theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field $k$ to be a smooth connected $k$-group in which every smooth connected affine normal $k$-subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension...

Let $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that all level $m$...

The cohomology of Nakajima’s varieties is known to carry a natural Weyl group action. Here this fact is established using the method of intersection cohomology, in analogy with the definition of Springer’s representations.

Let X be an algebraic toric variety with respect to an action of an algebraic torus S. Let Σ be the corresponding fan. The aim of this paper is to investigate open subsets of X with a good quotient by the (induced) action of a subtorus T ⊂ S. It turns out that it is enough to consider open S-invariant subsets of X with a good quotient by T. These subsets can be described by subfans of Σ. We give a description of such subfans and also a description of fans corresponding to quotient varieties. Moreover,...

Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we consider endomorphisms of ${\mathbb{P}}_1$ of degree $2$, defined over $K$, with good reduction outside $S$. We prove that there exist only finitely many such endomorphisms, up to conjugation by ${\mathrm{PGL}}_2\left(R_S\right)$, admitting a periodic point in ${\mathbb{P}}_1\left(K\right)$ of order $\>3$. Also, all but finitely many classes with a periodic point in ${\mathbb{P}}_1\left(K\right)$ of order $3$ are parametrized by an irreducible curve.

We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.

We study the Zariski closures of orbits of representations of quivers of type $A$, $D$ ou $E$. With the help of Lusztig’s canonical base, we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth.

Let $$𝕜$$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $$𝕜$$ . Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $$𝕜$$ is algebraically closed. In this paper we prove that $${\mathbb{P}}_{𝕜}^2/\phantom{{\mathbb{P}}_{𝕜}^{2}G}G$$ is rational for an arbitrary field $$𝕜$$ of characteristic zero.