ON ALGEBRAIC VOLUME DENSITY PROPERTY

Let X X be a complete Killing vector field on an n n -dimensional connected Riemannian manifold. Our main purpose is to show that if X X has as few as n n closed orbits which are located properly with respect to each other, then X X must have periodic flow. Together with a known result, this implies that periodicity of the flow characterizes those complete vector fields having all orbits closed which can be Killing with respect to some Riemannian metric on a connected manifold M M . We give a generalization of this characterization which applies to arbitrary complete vector fields on M M .

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G. Let $$ \mathcal{A} $$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $$ \mathcal{A} $$ -group schemes $$ \mathcal{P} $$ whose generic fibers $$ {{\mathcal{P}} \left/ {K} \right.} $$ coincide with G; these are known as parahoric group schemes. The special fiber $$ {{\mathcal{P}} \left/ {K} \right.} $$ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $$ {{\mathcal{P}} \left/ {K} \right.} $$ has a Levi factor, and that any two Levi factors of $$ {{\mathcal{P}} \left/ {K} \right.} $$ are geometrically conjugate.

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, that is, a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analog of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SOn defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if is Cayley for some r≥0. Here denotes the split r-dimensional k-torus. These notions were introduced in 2006 by Lemire, Popov, and Reichstein, who classified Cayley and stably Cayley simple groups over an algebraically closed field of characteristic zero. In this paper, we study reductive Cayley groups over an arbitrary field k of characteristic zero. Our main results are a criterion for a reductive group G to be stably Cayley, formulated in terms of its character lattice, and a classification of stably Cayley simple groups.

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

We solve the question about the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups over the algebraic closures of finite fields of positive characteristic.

The author proves that if is the algebra of regular functions of any smooth affine indecomposable algebraic variety () then it can be recovered from its Lie algebra of regular vector fields using a certain multilinear polynomial mapping. It is established that if, for some natural number , a finitely generated Lie algebra over an algebraically closed field () satisfies all identities of the Lie algebra of all derivations of the power series algebra in commuting variables, then contains a proper subalgebra of finite codimension; moreover, for any maximal ideal of , either or can be embedded in .

Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations. This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher sig-natures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence re-lated Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For re-ductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1

Degree 2 cohomological invariants of linear algebraic groups

Equivariant Poincaré Polynomials and Counting Points over Finite Fields

Torsion in linear algebraic groups over large algebraic fields

Let K be a differential field with algebraically closed field of constants CK . Let K P V ∞ be the (iterated) Picard-Vessiot closure of K. Let G be a linear differential algebraic group over K, and X a differential algebraic torsor for G over K. We prove that X ( K P V ∞ ) is Kolchin dense in X. In the special case that G is finite-dimensional we prove that X ( K P V ∞ ) = X ( K diff ) (where Kdiff is the differential closure of K). We also give close relationships between Picard-Vessiot extensions of K and the category of torsors for suitable finite-dimensional linear differential algebraic groups over K.

The systematic study of multiple structures, made necessary (and possible) by the notion of scheme introduced by Grothendieck, began with the papers [Fo], [Fe] and was continuated in [BF1] and [M1], [M2], [M3] (the last three use also ideeas from [PS]).In [BF1], [BF2], along the classification, up to multiplicity 4, of multiple locally complete intersection (lci for short) structures on a smooth curve embedded in a smooth threefold, general classes of multiple structures are introduced, the so-called "primitive" and "quasiprimitive" structures.The primitive ones are characterized by the fact that, locally, they are defined by equations of the type x n = 0, y = 0, z = 0, . . ., u = 0; the quasiprimitive ones are those which are generically primitive.The general study of these structures was continued by several authors, from which we mentionIn this paper we give the construction of a class of multiple locally complete intersection (lci for short) structures on a smooth algebraic variety as support.This class contains the lci structures defined locally by the "the next" monomial equations, namely those which are defined locally by equations of the form x n = 0, y 2 = 0, z = 0, . . ., u = 0.

Arithmetic purity of strong approximation for homogeneous spaces

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological obstructions to it) by incorporating an approximation condition in the Euclidean topology. We prove that the tight approximation property is a stable birational invariant, is compatible with fibrations, and satisfies descent under torsors of linear algebraic groups. Its validity for a number of rationally connected varieties follows. Some concrete consequences are: smooth loops in the real locus of a smooth compactification of a real linear algebraic group, or in a smooth cubic hypersurface of dimension≥2{\geq 2}, can be approximated by rational algebraic curves; homogeneous spaces of linear algebraic groups over the function field of a real curve satisfy weak approximation.