Equivariant models of spherical varieties. (English) Zbl 1440.14232
Let \(G\) be a connected semisimple group over an algebraically closed field \(k\) of characteristic 0. Let \(Y =G/H\) be a spherical homogeneous space of \(G\). Let \(Y'\) be a spherical embedding of \(Y\). Let \(k_0\) be a subfield of \(k\). Let \(G_0\) be a \(k_0\)-model (\(k_0\)-form) of \(G\). It is shown that if \(G_0\) is an inner form of a split group, and if the subgroup \(H\) of \(G\) is spherically closed, then \(Y\) admits a \(G_0\)-equivariant \(k_0\)-model. When the assumption that \(H\) is spherically closed is replaced by the stronger assumption that \(H\) coincides with its normalizer in \(G\), it is shown that \(Y\) and \(Y'\) admit compatible \(G_0\)-equivariant \(k_0\)-models, and these models are unique. The paper is very clearly and carefully written, and discusses in detail relations to other works in the area.
Reviewer: Yuval Z. Flicker (Ariel)
MathOverflow Questions:
Action of \(N(H)/H\) on the colors of a spherical homogeneous space \(G/H\)Is any spherical subgroup conjugate to a subgroup defined over a smaller algebraically closed field?
MSC:
| 14M27 | Compactifications; symmetric and spherical varieties |
| 32M10 | Homogeneous complex manifolds |
| 14M17 | Homogeneous spaces and generalizations |
Software:
MathOverflowReferences:
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