arXiv subjects
Publications and source records attributed to Mikhail Borovoi.
We introduce the notion of a quasi-connected reductive group over an arbitrary field to be an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group.
By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set H^1(R,G) of a quasi-connected reductive group G over the field R of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.
We classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was obtained by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E_8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology, first and second, to obtain the classification over R.
Let G be a linear algebraic group over the field of real numbers R, and let Y be a right homogeneous space of G. We wish to find a real point of Y or to prove that Y has no real points. We describe a method to do that, implicitly using second nonabelian Galois cohomology. Our method is suitable for computer-assisted calculations.
In this paper we classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was done by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology to obtain the classification over R. For the nilpotent orbits this is in principle rather straightforward and the main problem is to determine the first Galois cohomology sets of a long list of centralizers (we compute the centralizers using computer). For the semisimple and mixed orbits we develop new methods based on Galois cohomology, first and second. We also consider real theta-representations in general, and derive a number of results that are useful for the classification of their orbits.
Let U be a unipotent group over the field of complex numbers C, acting on a complex algebraic variety X. Assume that there exists a surjective morphism of complex algebraic varieties f: X --> Y whose fibres are orbits of U. We show that if X and Y are smooth and all orbits of U in X have the same dimension, then the induced map on C-points X(C) --> Y(C) is a homotopy equivalence. Moreover, if U, X, Y, and f are defined over the field of real numbers R, then the induced map on R-points X(R) --> Y(R) is surjective and induces homotopy equivalences on connected components.
Let $k_0$ be a field of characteristic $0$ with algebraic closure $k$. Let $G$ be a connected reductive $k$-group, and let $Y$ be a spherical variety over $k$ (a spherical homogeneous space or a spherical embedding). Let $G_0$ be a $k_0$-model ($k_0$-form) of $G$. We give necessary and sufficient conditions for the existence of a $G_0$-equivariant $k_0$-model of $Y$.
We study the equivariant real structures on complex horospherical varieties, generalizing classical results known for toric varieties and flag varieties. In particular, we obtain a necessary and sufficient condition for the existence of such real structures and determine the number of equivalence classes. We then apply our results to classify the equivariant real structures on smooth projective horospherical varieties of Picard rank 1.
For a connected semisimple group G over the field of real numbers R, using a method of Onishchik and Vinberg, we compute the first Galois cohomology set H^1(R,G) in terms of Kac labelings of the affine Dynkin diagram of G.
Let X be a homogeneous space of a quasi-trivial k-group G, with geometric stabilizer H, over a number field k. We prove that under certain conditions on the character group of H, certain algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation vanish, because the abelian groups where they take values vanish. When H is connected or abelian, these algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation are the only ones, so we prove the Hasse principle and weak approximation for X under certain conditions. As an application, we obtain new sufficient conditions for the Hasse principle and weak approximation for linear algebraic groups.
Let $G$ be a connected semisimple group over an algebraically closed field $k$ of characteristic 0. Let $Y=G/H$ be a spherical homogeneous space of $G$, and let $Y'$ be a spherical embedding of $Y$. Let $k_0$ be a subfield of $k$. Let $G_0$ be a $k_0$ -model ($k_0$-form) of $G$. We show that if $G_0$ is an inner form of a split group and if the subgroup $H$ of $G$ is spherically closed, then $Y$ admits a $G_0$-equivariant $k_0$-model. If we replace the assumption that $H$ is spherically closed by the stronger assumption that $H$ coincides with its normalizer in $G$, then $Y$ and $Y'$ admit compatible $G_0$-equivariant $k_0$-models, and these models are unique.
We prove a Hasse principle for binary direct summands of the Chow motive of a smooth projective quadric Q over a number field F. Besides, we show that such summands are twists of Rost motives. In the case when F has at most one real embedding we describe a complete motivic decomposition of Q.
Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field extension of $k$ of degree bounded independently of the subgroups. Moreover, if $k$ is a real number field, we show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over the field of real numbers $\mathbb{R}$ are actually conjugate over a finite real extension of $k$ of degree bounded independently of the subgroups.
Let k_0 be a field of characteristic 0, and let k be a fixed algebraic closure of k_0. Let G be an algebraic k-group, and let Y be a G-variety over k. Let G_0 be a k_0 -model (k_0 -form) of G. We ask whether Y admits a G_0 -equivariant k_0 -model Y_0 of Y. We assume that Y admits a G_q -equivariant k_0 -model Y_q, where G_q is an inner form of G_0. We give a Galois-cohomological criterion for the existence of a G_0 -equivariant k_0 -model Y_0 of Y. We apply this criterion to spherical homogeneous varieties Y=G/H.
Let G be a simply connected absolutely simple algebraic group defined over the field of real numbers R. Let H be a simply connected semisimple R-subgroup of G. We consider the homogeneous space X=G/H. We ask: How many connected components has X(R)? We give a method of answering this question. Our method is based on our solutions of generalized Reeder puzzles.
Let G be a connected, compact, semisimple algebraic group over the field of real numbers R. Using Kac diagrams, we describe combinatorially the first Galois cohomology sets H^1(R,H) for all inner forms H of G. As examples, we compute explicitly H^1 for all real forms of the simply connected group of type E_7 (which has been known since 2013) and for all real forms of half-spin groups of type D_{2k} (which seems to be new).
Let X=G/H be the quotient of a connected reductive algebraic C-group G defined over the field of complex numbers C by a finite subgroup H. We describe the topological fundamental group of the homogeneous space X, which is nonabelian when H is nonabelian. Further, we construct an example of a homogeneous space X and an automorphism s of C such that the topological fundamental groups of X and of the conjugate variety sX are not isomorphic.
Soit X un espace homogène d'un groupe algébrique linéaire connexe G sur C. Soit x un C-point de X. On désigne par H le stabilisateur de x dans G. On montre qu'on peut définir algébriquement le groupe fondamental topologique π_1(X(C),x), si ce groupe fondamental topologique est abélien. Si Pic(G)=0 et H est connexe ou abélien, on calcule π_1(X(C),x) en termes des groupes de caractères de G et H. En outre, si G et X sont définis sur un corps algébriquement clos de caractéristique p quelconque, on calcule la partie première à p du groupe fondamental étale de X en termes des groupes de caractères de G et H (si Pic(G)=0 et H est connexe). Let X be a homogeneous space of a connected linear algebraic group G defined over C. Let x be a C-point of X. We denote by H the stabilizer of x in G. We show that if the topological fundamental group π_1(X(C),x) is abelian, then it can be defined algebraically. If Pic(G)=0 and H is connected or abelian, we compute π_1(X(C),x) in terms of the character groups of G and H. Furthermore, when G and X are defined over an algebraically closed field of arbitrary characteristic p, we compute the prime-to-p étale fundamental group of X in terms of the character groups of G and H (if Pic(G)=0 and H is connected).