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.

  • $\begingroup$ Thank you, Friedrich! Is there a reference for this fact? (I need the projectivity in order to use Galois descent.) $\endgroup$ – Mikhail Borovoi Jun 22 '16 at 8:53
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    $\begingroup$ You could cite the first remark after the "Théorème" of Luna, D.: Toute variété magnifique est sphérique. Transform. Groups 1 (1996), no. 3, 249–258. That doesn't mean that there is a proof. It is clear that Luna had the argument above in mind. $\endgroup$ – Friedrich Knop Jun 22 '16 at 11:58
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    $\begingroup$ Roman Avdeev informed me that the argument above is written in the proof of Proposition 3.18 in his paper Strongly solvable spherical subgroups and their combinatorial invariants. $\endgroup$ – Mikhail Borovoi Jul 16 '16 at 11:47

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.

  • $\begingroup$ Welcome to Math Overflow, Roman! Note that I have a feeling that on this site, links are given only to papers that are freely accessible. Therefore, while the link to Losev's paper is perfectly OK, the link to Brion's paper seems problematic (even though there is no other link!). $\endgroup$ – Mikhail Borovoi Jul 16 '16 at 17:59
  • $\begingroup$ @Mikhael: While it's usually more helpful to link to sources which are freely accessible, that's not always possible (unless people have their own access via libraries). It changes over time. In any case, Elsevier has begun to allow free access to older papers such as the one here by Brion in J. Algebra. (But the more recent issues are still blocked to non-subscribers.) $\endgroup$ – Jim Humphreys Jul 16 '16 at 18:24
  • $\begingroup$ @Mikhail: Thanks for your hint concerning links to sources that are freely accessible, I'll keep this in mind! Actually, from my home computer I can freely access Brion's paper but cannot Losev's one. However the latter doesn't seem to be a problem since Losev's paper is on arXiv. $\endgroup$ – Roman Avdeev Jul 16 '16 at 19:42
  • $\begingroup$ @Roman: Right, I wrote hastily. Brion's paper is freely accessible, while Losev's one is not. I took the liberty to change the link to arXiv in your answer (I think the link in your comment is not sufficient). $\endgroup$ – Mikhail Borovoi Jul 16 '16 at 21:08

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