# Is the wonderful compactification of a spherical homogeneous variety always projective?

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 and Knop 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.

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.
Apart from the general argument mentioned by Friedrich in his answer, in the particular case $N_G(H) = H$ the projectivity of the wonderful compactification $X$ of $G/H$ can be seen directly. Namely, let $\mathfrak g$ and $\mathfrak h$ be the Lie algebras of $G$ and $H$, respectively. Then $X$ is isomorphic to the $G$-orbit closure $\overline{G \cdot [\mathfrak h]}$ in the Grassmannian $\mathrm{Gr}_{\dim \mathfrak h}(\mathfrak g)$. This result was proved by Losev in his paper Demazure embeddings are smooth from 2009. In fact, the projectivity of $X$ is deduced from the weaker fact due to Brion stating that $X$ is isomorphic to the normalization of $\overline{G \cdot [\mathfrak h]}$; see his paper Vers une généralisation des espaces symétriques from 1990.