Equivariant Models of Spherical Varieties

Borovoi, Mikhail

doi:10.1007/s00031-019-09531-w

Cited by 7 publications

(3 citation statements)

References 40 publications

(88 reference statements)

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“…Here, we discuss antiregular maps of complex affine varieties and antiregular automorphisms of complex algebraic groups. For the notion of a semilinear morphism of schemes over an arbitrary field see [10]. In this appendix, we write 𝑌(ℂ) (and not just 𝑌) for the set of ℂ-points of a ℂvariety 𝑌.…”

Section: Appendix: Antiregular Mapsmentioning

confidence: 99%

Computing Galois cohomology of a real linear algebraic group

Borovoi,

de Graaf

2024

Journal of London Math Soc

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Let be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers . We describe a method, implemented on computer, to find the first Galois cohomology set . The output is a list of 1‐cocycles in . Moreover, we describe an implemented algorithm that, given a 1‐cocycle , finds the cocycle in the computed list to which is equivalent, together with an element of realizing the equivalence.

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Section: Appendix: Antiregular Mapsmentioning

confidence: 99%

Computing Galois cohomology of a real linear algebraic group

Borovoi,

de Graaf

2024

Journal of London Math Soc

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“…• Equivariant real structures on spherical homogeneous spaces X 0 = G/H, under the extra assumption that Aut G (X 0 ) N G (H)/H is finite, have been studied by Akhiezer in [Akh15], by Akhiezer-Cupit-Foutou in [ACF14], by Cupit-Foutou in [CF15], by Borovoi in [Bor20], and by Snegirov in [Sne20].…”

Section: Real Structures On Spherical Varietiesmentioning

confidence: 99%

Real structures on complex algebraic varieties with a reductive group action

Terpereau¹

2022

Preprint

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This is a survey article devoted to the study of real structures on complex algebraic varieties endowed with a reductive group action. ContentsIntroduction 1 1. Real group structures on reductive groups 4 2. Equivariant real structures on G-varieties 6 3. Real structures on linear representations and nilpotent orbits 9 4. Real structures on varieties with a torus action 12 5. Real structures on spherical varieties 17 6. Real structures on almost homogeneous SL 2 -threefolds 21 7. Some open questions 24 References 25

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“…Let us mention that forms of spherical homogeneous spaces (see Definition 2.13) over an arbitrary base field of characteristic zero were studied by Borovoi and Gagliardi in [Bor20,BG]. The reader is referred to [MJT, § 3] and [BG,§ 11] for examples of spherical homogeneous spaces for which versions of Proposition A and Theorem B are applied to determine their (k, F )-forms.…”

Section: Proof Of Proposition Amentioning

confidence: 99%

Forms of almost homogeneous varieties over perfect fields

Moser-Jauslin¹,

Terpereau²

2020

Preprint

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We study the forms of the almost homogeneous varieties. More precisely, we first obtain criteria for the existence of forms in the homogeneous case, and then we extend the Luna-Vust theory over perfect fields to determine whether a given form of the dense open orbit of an almost homogeneous variety comes from a form of the whole variety. Finally, we apply our results to study the real forms of the complex almost homogeneous SL 2 -threefolds. ContentsIntroduction 1 Notation 5 1. Forms of homogeneous spaces 6 2. Luna-Vust theory over perfect fields 14 3. Real forms of complex almost homogeneous SL 2 -threefolds 21 Appendices 36 References 43The second-named author is supported by the ANR Project FIBALGA ANR-18-CE40-0003-01. This work received partial support from the French "Investissements d'Avenir" program and from project ISITE-BFC (contract ANR-lS-IDEX-OOOB). The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002). 1 2 LUCY MOSER-JAUSLIN AND RONAN TERPEREAUk and a G-variety X over k, we will fix a k-form F of G in the category of algebraic groups over k, and then consider the (k, F )-forms of X, i.e. the F -varieties Z over k such that Z k ≃ X as G-varieties over k. Any such form corresponds to an effective (G, ρ)-equivariant descent datum on X (here ρ refers to the decent datum on G associated with the k-form F ) through the map X → X/Γ.Two descent data on an algebraic group G are equivalent if they are conjugate by an element of Aut k (G), in which case they correspond to isomorphic k-forms of G. Two (G, ρ)-equivariant descent data on a G-variety X are equivalent if they are conjugate by an element of Aut G k (X), in which case they correspond to isomorphic (k, F )-forms of X.We can now state our mains results concerning the (k, F )-forms of arbitrary homogeneous spaces (Proposition A and Theorem B).Proposition A. ( § 1.4) Let G be a connected linear algebraic group over k, and let F be a k-form of G corresponding to the descent datum ρ on G. Let H ⊆ G be an algebraic subgroup. The homogeneous space X = G/H admits a (k, F )-form if and only if there exists a continuous map t : Γ → G(k) such thatIf (1)-( 2) are verified, then a (G, ρ)-equivariant descent datum on X is given byMoreover, if ρ 1 and ρ 2 are two strongly equivalent (i.e. conjugate by an inner automorphism of G) descent data on G, with corresponding k-forms F 1 and F 2 , then there is a bijection between the isomorphism classes of (k, F 1 )-forms and of (k, F 2 )forms of X.Remark 0.1. The existence of a (k, F )-form of X = G/H does not depend on the choice of a base point of X. Indeed, if the two conditions in Proposition A hold and H ′ = sHs −1 for some s ∈ G(k), then a (G, ρ)-equivariant descent datum on G/H ′ is given by µRemark 0.2. If ρ 1 and ρ 2 are two equivalent descent data on G, but not strongly equivalent, with corresponding k-forms F 1 and F 2 , then the existence of a (k, F 1 )form for X does not imply the existence of a (k, F 2 )-form for X; see Example 1.11.Theorem B. ( § 1.4) We keep the notation of Propo...

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Equivariant Models of Spherical Varieties

Cited by 7 publications

References 40 publications

Computing Galois cohomology of a real linear algebraic group

Computing Galois cohomology of a real linear algebraic group

Real structures on complex algebraic varieties with a reductive group action

Forms of almost homogeneous varieties over perfect fields

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