Let $G/H$ be a spherical homogeneous variety, where $G$ is a complex semisimple group. Assume that the subgroup $H$ is self-normalizing, i.e., $\mathcal{N}\_G(H)=H$. Then by results of [Brion and Pauer](http://link.springer.com/article/10.1007%2FBF02564447)
and [Knop](http://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00179-8/S0894-0347-96-00179-8.pdf)
there exists a wonderful compactification of $G/H$. Is this wonderful compactification always projective, or in some cases it can be complete, but not projective?


Feel free to vote to close this elementary question.



Is the wonderful compactification of a spherical homogeneous variety always projective?
The wonderful compactification is always projective. One way to see is to use a theorem of Sumihiro which says that a normal $G$-variety is covered by $G$-invariant quasiprojective open subsets. Since a wonderful variety $X$ has only one closed orbit $Y$ the only $G$-invariant open subset meeting $Y$ is $X$ itself.


