# Spherical and Wonderful varieties

A spherical variety is a normal variety $X$ together with an action of a connected reductive affine algebraic group $G$, a Borel subgroup $B\subset G$, and a base point $x_0\in X$ such that the $B$-orbit of $x_0$ in $X$ is a dense open subset of $X$.

A wonderful variety is a smooth complete variety $X$ with the action of a semisimple simply connected group $G$ such that there is a point $x_0\in X$ with open $G$ orbit and such that the complement $X\setminus G\cdot x_0$ is a union of prime divisors $E_1,\cdots, E_t$ having simple normal crossing, and such that the closures of the $G$-orbits in $X$ are the intersections $\bigcap_{i\in I}E_i$ where $I$ is a subset of $\{1,\dots, t\}$.

Now, fix a connected reductive affine algebraic group $G$ and a Borel subgroup $B\subset G$. Could there exist two non isomorphic smooth complete varieties that are spherical with respect to $(G,B)$ and wonderful with respect to $G$ ?

• wouldn't already G/P for two different parabolic subgroups P fit the bill? They are already complete, so there is no condition on boundary divisors. There are some easy non-isomorphic examples, e.g. Grassmannians Gr(k,n) for different k. Nov 14, 2017 at 10:21
• @Misa: I guess your question is posed over an algebraically closed field of characteristic 0 such as $\mathbb{C}$? (Does the characteristic matter?) Nov 15, 2017 at 0:29
• Yes, $X$ is a variety over an algebraically closed field of characteristic 0.
– user114666
Dec 16, 2017 at 17:32

The only groups which act on only one wonderful variety are tori (with $X$ being a point). All other admit at least $X=G/B$ and $X=G/G$.
If one fixes the open $G$-orbit then there is at most one wonderful completion (Luna-Vust, Luna).
It is known that the number of wonderful varieties for $G$ is finite (work of Alexeev-Brion). By now, they are all classified (work of mostly Luna, Losev, and Bravi-Pezzini). The result is way too complicated to describe it here. Bravi and Luna have written a nice survey on the case where $G=F_4$.