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 $G$ ?



Spherical and Wonderful varieties
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 $G=F\_4$.


