Linear algebraic group

Abstract: A linear algebraic group over an algebraically closed field k is a subgroup of a group GL n (k) of invertible n × n-matrices with entries in k, whose elements are precisely the solutions of a set of polynomial equations in the matrix coordinates. The present article contains a review of the theory of linear algebraic groups.

Abstract: Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those of the arithmetic groups. If G is a linear algebraic group defined over Q, X is the symmetric space associated to G and A is an arithmetic subgroup of G, then A is virtually torsion free and acts properly discontinuously on X. The rational homology of A is the same as that of X/A. Furthermore there is a "bordification" of X ([BS]) to a manifold with corners 3~ and an extension of the action of A to a properly discontinuous action on Jf so that the quotient X / A is compact. The boundary of Jf is homotopy equivalent to a wedge of spheres, say of dimension d, and the virtual cohomological dimension of A is n d + 1, where n is the dimension of X. In the case of the mapping class group there is no analog for G, Ivanov (unpublished) has proven that F is not arithmetic. Nevertheless, F acts properly discontinuously on Teichmiiller space r which is homeomorphic to Euclidean space (of dimension 6 g 6 + 2 s ) ; 3"will play the role of the symmetric space. The quotient of .9by F is the moduli space of curves whose rational homology is then identified with that of E Harvey [Har] has constructed a Borel-Serre bordification J of 3-by analytic methods. In the case where F has punctures, we will build g by a different, combinatorial method. In addition, we will explicitly describe inside Y a cell complex Y of dimension 4 g 4 + s onto which J may be F-equivariantly retracted, thus establishing an analog of the constructions for SL n by Serre, Soul6 ([Sol) and Ash ([A]). This complex will be of the lowest possible dimension because we will use J= to prove our main result:

Abstract: In this article we present a fundamental result due to Sumihiro. It states that every normal G-variety X, where G is a connected linear algebraic group, is locally isomorphic to a quasi-projective G-variety, i.e., to a G-stable subvariety of the projective space P n with a linear G-action (Theorem 1.1). The central tools for the proof are G-linearization of line bundles (§2) and some properties of the Picard group of a linear algebraic group (§4).

Abstract: In my previous papers [2], [4], I have given a method of compactifying the quotient spaces of the generalized upper half plane with respect to Siegel's modular group and, more generally, to any group commensurable with Siegel's modular group. Now a similar problem can be considered in a more general situation as follows. Let G be a semi-simple linear algebraic group defined over Q, and let G4, GR be the groups consisting of points in G rational over Q, R, respectively. GR is a semisimple Lie group with a finite number of connected components. Let K be a maximal compact subgroup of GR and let S = K\GR be the associated (not necessarily connected) symmetric space. Let furthermore Gz be the "group of units" in G, i.e., the group consisting of all elements in G whose coefficients (together with those of its inverse) are rational integers. Then Gz is a discrete subgroup of GR, whose commensurable class is uniquely determined, independently of the choice of the matrix expression of G. Let F be any subgroup of Gq commensurable with Gz. Then one may ask the possibility of constructing a reasonable compactification of the quotient space S/r. To approach this problem, it will be convenient to use a "fundamental set" for F, which has been constructed by Weil [7], by means of the reduction theory, in the case where G is a group of automorphisms of a semi-simple associative algebra over Q with or without involution. Moreover he has also shown in [6] that these cases cover all semi-simple linear algebraic groups without center, of classical type, over Q, with few exceptions. The purpose of this paper is to give some results on the above problem in the case treated by Weil [7]. The outline of the paper is as follows. We shall recall in ? 1 the main result of our previous paper [3], on which our whole construction will be based, concerning a general method of compactification of a symmetric space S by means of an irreducible, faithful projective representation p of the corresponding Lie group; we give also a trivial generalization of it to the non-connected case. In ? 2 we shall give a general condition (the condition (D)) for a discontinuous group r operating on S and for a fundamental set f2 for F, which enables us to construct a suitable compacti-