Mathematics > Algebraic Geometry
[Submitted on 4 May 2018 (v1), last revised 19 Aug 2019 (this version, v4)]
Title:Spherical varieties over large fields
View PDFAbstract:Let k_0 be a field of characteristic 0, k its algebraic closure, G a connected reductive group defined over k. Let H\subset G be a spherical subgroup. We assume that k_0 is a large field, for example, k_0 is either the field R of real numbers or a p-adic field. Let G_0 be a quasi-split k_0-form of G. We show that if H has self-normalizing normalizer, and Gal(k/k_0) preserves the combinatorial invariants of G/H, then H is conjugate to a subgroup defined over k_0, and hence, the G-variety G/H admits a G_0-equivariant k_0-form. In the case when G_0 is not assumed to be quasi-split, we give a necessary and sufficient Galois-cohomological condition for the existence of a G_0-equivariant k_0-form of G/H.
Submission history
From: Stephan Snegirov [view email][v1] Fri, 4 May 2018 17:36:00 UTC (14 KB)
[v2] Wed, 9 May 2018 18:22:41 UTC (14 KB)
[v3] Tue, 2 Oct 2018 00:27:41 UTC (14 KB)
[v4] Mon, 19 Aug 2019 17:45:43 UTC (22 KB)
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