Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Forms of almost homogeneous varieties Lucy Moser-Jauslin Universit´ e de Bourgogne, France QFLAG Seminar November 9, 2020 (In collaboration with Ronan Terpereau) https ://arxiv.org/abs/2008.05197 Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Let k be a perfect field with algebraic closure k , and let G be a connected reductive k -group. Definition A normal G -variety is almost homogeneous if it has an open dense orbit. Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Let k be a perfect field with algebraic closure k , and let G be a connected reductive k -group. Definition A normal G -variety is almost homogeneous if it has an open dense orbit. Problem Given an almost homogeneous G -variety X , find all k -forms on X which are compatible with a given k -form F of G . Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Definition A k-form of the algebraic group G is an algebraic group F over k together with an isomorphism F k = F × Spec ( k ) Spec ( k ) ≃ G of algebraic groups. Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Definition A k-form of the algebraic group G is an algebraic group F over k together with an isomorphism F k = F × Spec ( k ) Spec ( k ) ≃ G of algebraic groups. A ( k , F ) -form of a G -variety X is an F -variety Z together with an isomorphism Z k = Z × Spec ( k ) Spec ( k ) ≃ X such that the action of G on X is defined by the isomorphism F k ≃ G . Lucy Moser-Jauslin Forms of almost homogeneous varieties
� � � Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Let Γ = Gal ( k / k ). Definition A descent datum on G is a continuous Γ-action ρ : Γ × G → G such that, for each γ ∈ Γ, there is a commutative diagram ρ γ G G ( γ ∗ ) − 1 � Spec ( k ) Spec ( k ) where ρ γ = ρ ( γ, − ) ∈ Aut k ( G ) is the automorphism induced by ρ . Lucy Moser-Jauslin Forms of almost homogeneous varieties
� � � Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Let Γ = Gal ( k / k ). Definition A descent datum on G is a continuous Γ-action ρ : Γ × G → G such that, for each γ ∈ Γ, there is a commutative diagram ρ γ G G ( γ ∗ ) − 1 � Spec ( k ) Spec ( k ) where ρ γ = ρ ( γ, − ) ∈ Aut k ( G ) is the automorphism induced by ρ . Two descent data ρ 1 and ρ 2 on G are equivalent if there exists ψ ∈ Aut k ( G ) such that ∀ γ ∈ Γ , ρ 2 ,γ = ψ ◦ ρ 1 ,γ ◦ ψ − 1 . Lucy Moser-Jauslin Forms of almost homogeneous varieties
� � � Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Definition Let X be a G -variety, and ρ be a descent datum on G . A (G , ρ ) -descent datum on X is a continuous Γ-action µ : Γ × X → X such that : µ γ ( g · x ) = ρ γ ( g ) · µ γ ( x ) for all γ ∈ Γ, g ∈ G and x ∈ X ; and for each γ ∈ Γ, there is a commutative diagram µ γ X X ( γ ∗ ) − 1 � Spec ( k ) Spec ( k ) where µ γ is a scheme automorphism over Spec ( k ). Lucy Moser-Jauslin Forms of almost homogeneous varieties
� � � Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Definition Let X be a G -variety, and ρ be a descent datum on G . A (G , ρ ) -descent datum on X is a continuous Γ-action µ : Γ × X → X such that : µ γ ( g · x ) = ρ γ ( g ) · µ γ ( x ) for all γ ∈ Γ, g ∈ G and x ∈ X ; and for each γ ∈ Γ, there is a commutative diagram µ γ X X ( γ ∗ ) − 1 � Spec ( k ) Spec ( k ) where µ γ is a scheme automorphism over Spec ( k ). Two ( G , ρ )-descent data µ 1 and µ 2 on X are equivalent if there exists ψ ∈ Aut G ( X ) such that for all γ ∈ Γ : µ 2 ,γ = ψ ◦ µ 1 ,γ ◦ ψ − 1 . Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties If ρ is a descent datum on G , then G / Γ is a k -form of G . Also, all k -forms of G arise as the quotient of G be the action of Γ from a descent datum. Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties If ρ is a descent datum on G , then G / Γ is a k -form of G . Also, all k -forms of G arise as the quotient of G be the action of Γ from a descent datum. If µ is a ( G , ρ )-descent datum on X such that X is covered by Γ-stable affine open subsets, then X / Γ is a k -form of G . Also, all k -forms of X arise as the quotient of X be the action of Γ from a descent datum of this form. Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Related works Huruguen (2011-2012) : Toric varieties, spherical varieties Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Related works Huruguen (2011-2012) : Toric varieties, spherical varieties Wedhorn (2018) Spherical varieties Cupit-Foutou, Akheizer (2012) (Real forms of certain spherical varieties) Cupit-Foutou, Timashev (2017,2019) (Real orbits of complex spherical homogeneous spaces) Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Related works Huruguen (2011-2012) : Toric varieties, spherical varieties Wedhorn (2018) Spherical varieties Cupit-Foutou, Akheizer (2012) (Real forms of certain spherical varieties) Cupit-Foutou, Timashev (2017,2019) (Real orbits of complex spherical homogeneous spaces) Borovoi-Gagliardi (2018-20) (General theory of forms of G -varieties, with more specific results for spherical varieties) Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Related works Huruguen (2011-2012) : Toric varieties, spherical varieties Wedhorn (2018) Spherical varieties Cupit-Foutou, Akheizer (2012) (Real forms of certain spherical varieties) Cupit-Foutou, Timashev (2017,2019) (Real orbits of complex spherical homogeneous spaces) Borovoi-Gagliardi (2018-20) (General theory of forms of G -varieties, with more specific results for spherical varieties) MJ-T (2018-19) : (horospherical real case (with Borovoi), symmetric real varieties) Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Overall method Let X be an almost homogeneous G -variety with open orbit ∼ = G / H . Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Overall method Let X be an almost homogeneous G -variety with open orbit ∼ = G / H . Find all descent data ρ on G . Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Overall method Let X be an almost homogeneous G -variety with open orbit ∼ = G / H . Find all descent data ρ on G . For each ρ , find all ( G , ρ )-descent data µ on G / H . Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Overall method Let X be an almost homogeneous G -variety with open orbit ∼ = G / H . Find all descent data ρ on G . For each ρ , find all ( G , ρ )-descent data µ on G / H . Determine which µ extend to a descent datum on X . Lucy Moser-Jauslin Forms of almost homogeneous varieties
Main question k -forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL 2 ( C ) -varieties Let G / H be a homogeneous variety, and ρ a descent datum of G . Existence of a ( G , ρ )-descent datum G / H admits a ( G , ρ )-descent datum if and only if there exists a continuous map t : Γ → G such that ρ γ ( H ) = t γ Ht − 1 for all γ ∈ Γ ; and γ t γ 1 γ 2 ∈ ρ γ 1 ( t γ 2 ) t γ 1 H for all γ 1 , γ 2 ∈ Γ. Lucy Moser-Jauslin Forms of almost homogeneous varieties
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