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.
 A: 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.
A: 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.
