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.

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.

Le théorème de Borel-Weil-Bott décrit la cohomologie des fibrés en droites sur les variétés de drapeaux. On généralise ici ce théorème à une plus grande classe de variétés projectives : les variétés magnifiques de rang minimal.

De Concini and Procesi have defined the wonderful compactification $\overline{X}$ of a symmetric space $X=G/G^{\sigma }$ where $G$ is a complex semisimple adjoint group and $G^{\sigma }$ the subgroup of fixed points of $G$ by an involution $\sigma$. 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...

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 $\mathsf{A}$ and the prescribed weight monoid is that of a spherical module.

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 $K{\left(X\right)}_{\mathbb{Q}}$ where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure constants of...

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 $\mathbb{P}\left(\text{End}\right(V\left)\right)$. We determine for which $V$ the variety $X$ is normal and for which $V$ is smooth.

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.

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.

These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel${}^{\text{'}}$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.