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Classification of spherical varieties

Paolo Bravi (2010)

Les cours du CIRM

We give a short introduction to the problem of classification of spherical varieties, by presenting the Luna conjecture about the classification of wonderful varieties and illustrating some of the related currently known results.

Classification of strict wonderful varieties

Paolo Bravi, Stéphanie Cupit-Foutou (2010)

Annales de l’institut Fourier

In the setting of strict wonderful varieties we prove Luna’s conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that primitive strict wonderful varieties are mostly obtained from symmetric spaces, spherical nilpotent orbits and model spaces. To make the paper as self-contained as possible, we also gather some known results on these families and more generally on wonderful varieties.

Equations of some wonderful compactifications

Pascal Hivert (2011)

Annales de l’institut Fourier

De Concini and Procesi have defined the wonderful compactification X ¯ of a symmetric space X = G / G σ where G is a complex semisimple adjoint group and G σ the subgroup of fixed points of G by an involution σ . It is a closed subvariety of a Grassmannian of the Lie algebra 𝔤 of G . In this paper we prove that, when the rank of X is equal to the rank of G , the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form w on 𝔤 vanishes on the ( - 1 ) -eigenspace...

Equivariant degenerations of spherical modules for groups of type A

Stavros Argyrios Papadakis, Bart Van Steirteghem (2012)

Annales de l’institut Fourier

V. Alexeev and M. Brion introduced, for a given a complex reductive group, a moduli scheme of affine spherical varieties with prescribed weight monoid. We provide new examples of this moduli scheme by proving that it is an affine space when the given group is of type A and the prescribed weight monoid is that of a spherical module.

Equivariant K-theory of flag varieties revisited and related results

V. Uma (2013)

Colloquium Mathematicae

We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring K T ( G / B ) of the flag variety G/B. We do this by lifting the classes of the structure sheaves of Schubert varieties in K T ( G / B ) to R(T) ⊗ R(T), where R(T) denotes the representation ring of the torus T. We further apply our results to describe the multiplicative structure constants of K ( X ) where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure constants of...

Normality and non-normality of group compactifications in simple projective spaces

Paolo Bravi, Jacopo Gandini, Andrea Maffei, Alessandro Ruzzi (2011)

Annales de l’institut Fourier

Given an irreducible representation V of a complex simply connected semisimple algebraic group G we consider the closure X of the image of G in ( End ( V ) ) . We determine for which V the variety X is normal and for which V is smooth.

Spherical roots of spherical varieties

Friedrich Knop (2014)

Annales de l’institut Fourier

Brion proved that the valuation cone of a complex spherical variety is a fundamental domain for a finite reflection group, called the little Weyl group. The principal goal of this paper is to generalize this theorem to fields of characteristic unequal to 2. We also prove a weaker version which holds in characteristic 2, as well. Our main tool is a generalization of Akhiezer’s classification of spherical varieties of rank 1.

Spherical varieties and Wahl’s conjecture

Nicolas Perrin (2014)

Annales de l’institut Fourier

Using the theory of spherical varieties, we give a type independent very short proof of Wahl’s conjecture for cominuscule homogeneous varieties for all primes different from 2.

Toric and tropical compactifications of hyperplane complements

Graham Denham (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel ' fand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.

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