These notes contain an introduction to the theory of spherical and wonderful varieties. We describe the Luna-Vust theory of embeddings of spherical homogeneous spaces, and explain how wonderful varieties fit in the theory.
Dans cet article, nous définissons une catégorie des motifs sur une catégorie monoïdale symétrique vérifiant certaines hypothèses. Le rôle des espaces sur est joué par les monoïdes (non necessairement commutatifs) dans . Pour définir les morphismes dans , nous utilisons des classes dans les groupes d’homologie cyclique bivariante. Le but est de montrer que les opérateurs de périodicité de Connes induisent des morphismes dans , où est le motif de Tate dans .
We study liftings or deformations of -modules ( is the ring of differential operators from EGA IV) from positive characteristic to characteristic zero using ideas of Matzat and Berthelot’s theory of arithmetic -modules. We pay special attention to the growth of the differential Galois group of the liftings. We also apply formal deformation theory (following Schlessinger and Mazur) to analyze the space of all liftings of a given -module in positive characteristic. At the end we compare the problems...
Let G be one of the groups SLn(ℂ), Sp2n (ℂ), SOm(ℂ), Om(ℂ), or G 2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙN is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.
Let be a connected complex reductive group where is a finite-dimensional complex vector space. Let and let . Following Raïs we say that the orbit is characteristic for if the identity component of is . If is semisimple, we say that is semi-characteristic for if the identity component of is an extension of by a torus. We classify the -orbits which are not (semi)-characteristic in many cases.
The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a Hermitean symmetric quotient of type is computed. The main ingredient is a local version of Borcherds’ automorphic products. The local obstructions for a Heegner divisor to be principal are given by certain theta series with harmonic coefficients. Sometimes they generate Borcherds’ space of global obstructions. In these particular cases we obtain a simple proof of a result due to the first author: Suppose...