Forms of almost homogeneous varieties Lucy Moser-Jauslin Universit - - PowerPoint PPT Presentation

Main question k-forms and descent data Homogeneous case Luna-Vust theory Real forms of almost homogeneous SL2(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 SL2(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

• rbit.

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 SL2(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

• rbit.

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 SL2(C)-varieties

Definition A k-form of the algebraic group G is an algebraic group F

• ver 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 SL2(C)-varieties

Definition A k-form of the algebraic group G is an algebraic group F

• ver 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 Zk = 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 SL2(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 G

ργ

• G
• Spec(k)

(γ∗)−1 Spec(k)

where ργ = ρ(γ, −) ∈ Autk(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 SL2(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 G

ργ

• G
• Spec(k)

(γ∗)−1 Spec(k)

where ργ = ρ(γ, −) ∈ Autk(G) is the automorphism induced by ρ. Two descent data ρ1 and ρ2 on G are equivalent if there exists ψ ∈ Autk(G) such that ∀γ ∈ Γ, ρ2,γ = ψ ◦ ρ1,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 SL2(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 :

µγ(g · x) = ργ(g) · µγ(x) for all γ ∈ Γ, g ∈ G and x ∈ X ; and for each γ ∈ Γ, there is a commutative diagram X

µγ

• X
• Spec(k)

(γ∗)−1 Spec(k)

where µγ 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 SL2(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 :

µγ(g · x) = ργ(g) · µγ(x) for all γ ∈ Γ, g ∈ G and x ∈ X ; and for each γ ∈ Γ, there is a commutative diagram X

µγ

• X
• Spec(k)

(γ∗)−1 Spec(k)

where µγ is a scheme automorphism over Spec(k).

Two (G, ρ)-descent data µ1 and µ2 on X are equivalent if there exists ψ ∈ AutG(X) such that for all γ ∈ Γ : µ2,γ = ψ ◦ µ1,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 SL2(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

• f 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 SL2(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

• f 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 SL2(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 SL2(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 SL2(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 SL2(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 SL2(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 SL2(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 SL2(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 SL2(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 SL2(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 ργ(H) = tγHt−1

γ

for all γ ∈ Γ ; and tγ1γ2 ∈ ργ1(tγ2)tγ1H for all γ1, γ2 ∈ Γ.

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 SL2(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 ργ(H) = tγHt−1

γ

for all γ ∈ Γ ; and tγ1γ2 ∈ ργ1(tγ2)tγ1H for all γ1, γ2 ∈ Γ. µγ(gH) = ργ(g)tγH.

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 SL2(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 ργ(H) = tγHt−1

γ

for all γ ∈ Γ ; and tγ1γ2 ∈ ργ1(tγ2)tγ1H for all γ1, γ2 ∈ Γ. µγ(gH) = ργ(g)tγH. Example If Γ stabilizes H, then G/H admits a (G, ρ)-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 SL2(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 ργ(H) = tγHt−1

γ

for all γ ∈ Γ ; and tγ1γ2 ∈ ργ1(tγ2)tγ1H for all γ1, γ2 ∈ Γ. µγ(gH) = ργ(g)tγH. Example If Γ stabilizes H, then G/H admits a (G, ρ)-descent datum. (If there exists at least one, then the set of all equivalence classes

• f (G, ρ)-descent data is given by H1(Γ, NG(H)/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 SL2(C)-varieties

G/H is always quasi-projective.

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 SL2(C)-varieties

G/H is always quasi-projective. Thus each (G, ρ)-descent datum gives rise to an (k, F)-form of G/H, where F = G/Γ, with the action of Γ on G 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 SL2(C)-varieties

Normal G-varieties, where G is a reductive connected k-group. Almost homogeneous varieties An almost homogeneous G-variety is a G-variety with an open

• rbit together with an an isomorphism G/H → the open 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 SL2(C)-varieties

Normal G-varieties, where G is a reductive connected k-group. Almost homogeneous varieties An almost homogeneous G-variety is a G-variety with an open

• rbit together with an an isomorphism G/H → the open orbit.

Examples : 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 SL2(C)-varieties

Normal G-varieties, where G is a reductive connected k-group. Almost homogeneous varieties An almost homogeneous G-variety is a G-variety with an open

• rbit together with an an isomorphism G/H → the open orbit.

Examples : Toric varieties, spherical varieties,... Complexity The complexity of a G-variety X is the minimal codimension of a B-orbit in X, where B is a Borel subgroup 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 SL2(C)-varieties

Normal G-varieties, where G is a reductive connected k-group. Almost homogeneous varieties An almost homogeneous G-variety is a G-variety with an open

• rbit together with an an isomorphism G/H → the open orbit.

Examples : Toric varieties, spherical varieties,... Complexity The complexity of a G-variety X is the minimal codimension of a B-orbit in X, where B is a Borel subgroup of G. Complexity 0 = 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 SL2(C)-varieties

Normal G-varieties, where G is a reductive connected k-group. Almost homogeneous varieties An almost homogeneous G-variety is a G-variety with an open

• rbit together with an an isomorphism G/H → the open orbit.

Examples : Toric varieties, spherical varieties,... Complexity The complexity of a G-variety X is the minimal codimension of a B-orbit in X, where B is a Borel subgroup of G. Complexity 0 = Spherical varieties Example of complexity 1 almost homogeneous vari´ eties : the 3-dimensional SL2(k) case.

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 SL2(C)-varieties

Luna-Vust (1983) : A combinatorial classification of almost homogeneous varieties over an algebraically closed field of characteristic zero.

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 SL2(C)-varieties

Luna-Vust (1983) : A combinatorial classification of almost homogeneous varieties over an algebraically closed field of characteristic zero. Timashev (1997, 2011) : Generalization to any algebraically closed field, (and to more general G-varieties) This classification is effective when the complexity is 0 or 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 SL2(C)-varieties

X=almost homogeneous variety with open orbit G/H. Fix a Borel subgroup B of G. Fact : X is covered by B-stable affine open subvarieties (Sumihiro)

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 SL2(C)-varieties

X=almost homogeneous variety with open orbit G/H. Fix a Borel subgroup B of G. Fact : X is covered by B-stable affine open subvarieties (Sumihiro) Let K be the field K = k(G/H) = k(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 SL2(C)-varieties

X=almost homogeneous variety with open orbit G/H. Fix a Borel subgroup B of G. Fact : X is covered by B-stable affine open subvarieties (Sumihiro) Let K be the field K = k(G/H) = k(G)H. VG = {G-invariant geometric valuations on 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 SL2(C)-varieties

X=almost homogeneous variety with open orbit G/H. Fix a Borel subgroup B of G. Fact : X is covered by B-stable affine open subvarieties (Sumihiro) Let K be the field K = k(G/H) = k(G)H. VG = {G-invariant geometric valuations on K}. v ∈ VG is determined by its values on B-semi-invariants.

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 SL2(C)-varieties

X=almost homogeneous variety with open orbit G/H. Fix a Borel subgroup B of G. Fact : X is covered by B-stable affine open subvarieties (Sumihiro) Let K be the field K = k(G/H) = k(G)H. VG = {G-invariant geometric valuations on K}. v ∈ VG is determined by its values on B-semi-invariants. The colors : DB = {B-stable prime divisors of 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 SL2(C)-varieties

X=almost homogeneous variety with open orbit G/H. Fix a Borel subgroup B of G. Fact : X is covered by B-stable affine open subvarieties (Sumihiro) Let K be the field K = k(G/H) = k(G)H. VG = {G-invariant geometric valuations on K}. v ∈ VG is determined by its values on B-semi-invariants. The colors : DB = {B-stable prime divisors of G/H}. (VG, DB) is the colored equipment of G/H (or of 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 SL2(C)-varieties

X=almost homogeneous variety with open orbit G/H. Fix a Borel subgroup B of G. Fact : X is covered by B-stable affine open subvarieties (Sumihiro) Let K be the field K = k(G/H) = k(G)H. VG = {G-invariant geometric valuations on K}. v ∈ VG is determined by its values on B-semi-invariants. The colors : DB = {B-stable prime divisors of G/H}. (VG, DB) is the colored equipment of G/H (or of X). For each G-orbit Y of X, consider the pair (VG

Y , DB Y ), where

VG

Y = {vD ∈ VG|Y ⊂ D where D is a G-stable prime divisor ofX} ;

DB

Y = {D ∈ DB|Y ⊂ D}.

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 SL2(C)-varieties

colored data

The colored data of X is the collection of pairs (VG

Y , DB Y ) for all

G-orbits Y of X. The colored data determines 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 SL2(C)-varieties

Γ-action on colored equipment.

(Huruguen) Fix a descent datum ρ on G and a (G, ρ)-descent datum µ on G/H. K = k(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 SL2(C)-varieties

Γ-action on colored equipment.

(Huruguen) Fix a descent datum ρ on G and a (G, ρ)-descent datum µ on G/H. K = k(G/H). Action of Γ on K : (γ · f )(x) = γ(f (µγ−1(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 SL2(C)-varieties

Γ-action on colored equipment.

(Huruguen) Fix a descent datum ρ on G and a (G, ρ)-descent datum µ on G/H. K = k(G/H). Action of Γ on K : (γ · f )(x) = γ(f (µγ−1(x))). Action on VG : (γ · v)(f ) = v(γ−1 · f ).

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 SL2(C)-varieties

Γ-action on colored equipment.

(Huruguen) Fix a descent datum ρ on G and a (G, ρ)-descent datum µ on G/H. K = k(G/H). Action of Γ on K : (γ · f )(x) = γ(f (µγ−1(x))). Action on VG : (γ · v)(f ) = v(γ−1 · f ). Action on DB : for each γ ∈ Γ, there exists eγ ∈ G such that ργ(B) = eγBe−1

γ .

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 SL2(C)-varieties

Γ-action on colored equipment.

(Huruguen) Fix a descent datum ρ on G and a (G, ρ)-descent datum µ on G/H. K = k(G/H). Action of Γ on K : (γ · f )(x) = γ(f (µγ−1(x))). Action on VG : (γ · v)(f ) = v(γ−1 · f ). Action on DB : for each γ ∈ Γ, there exists eγ ∈ G such that ργ(B) = eγBe−1

γ .

vγ·D(f ) = vD(γ−1 · (eγf )).

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 SL2(C)-varieties

Theorem Let X be an almost homogeneous G-space, with i : G/H → X a bijection onto the open orbit. Let ρ be a descent datum on G, and µ a (G, ρ)-descent datum on G/H. Then µ extends to a unique descent datum on X if and only if the colored data of X is stable under the action of Γ. If it exists, then X/Γ is a form of X if and only if X is covered by

• pen Γ-stable affine (or quasi-projective) subsets.

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 SL2(C)-varieties

Special case : k = R , k = C, G = SL2(C). Γ = {1, γ}. A descent datum is given by an anti-regular involution. G has two real forms : SL2(R) and SU2. The two descent data : σs(g) = g and σc(g) =t g −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 SL2(C)-varieties

Special case : k = R , k = C, G = SL2(C). Γ = {1, γ}. A descent datum is given by an anti-regular involution. G has two real forms : SL2(R) and SU2. The two descent data : σs(g) = g and σc(g) =t g −1. H=Finite subgroup (type An, Dn, E6, E7 or E8) Up to conjugacy, we can assume that σ(H) = 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 SL2(C)-varieties

Special case : k = R , k = C, G = SL2(C). Γ = {1, γ}. A descent datum is given by an anti-regular involution. G has two real forms : SL2(R) and SU2. The two descent data : σs(g) = g and σc(g) =t g −1. H=Finite subgroup (type An, Dn, E6, E7 or E8) Up to conjugacy, we can assume that σ(H) = H. Equivalence classes of real forms on G/H : H1(Γ, NG(H)/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 SL2(C)-varieties

The colored equipment

B = {

• ∗

∗ ∗

• }.

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 SL2(C)-varieties

The colored equipment

B = {

• ∗

∗ ∗

• }.

DB ∼ = B\G/H ∼ = P1

C/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 SL2(C)-varieties

When H = {Id} : For D ∈ DB, choose fD ∈ C[G] with zero set = D.

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 SL2(C)-varieties

When H = {Id} : For D ∈ DB, choose fD ∈ C[G] with zero set = D. fD(

• x

y z w

• ) = αx + βy.

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 SL2(C)-varieties

When H = {Id} : For D ∈ DB, choose fD ∈ C[G] with zero set = D. fD(

• x

y z w

• ) = αx + βy.

The set VG : Determined by values on {fD}, D ∈ DB.

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 SL2(C)-varieties

When H = {Id} : For D ∈ DB, choose fD ∈ C[G] with zero set = D. fD(

• x

y z w

• ) = αx + βy.

The set VG : Determined by values on {fD}, D ∈ DB. v ∈ VG : v(fD0) = r, with r ∈ [−1, 1] ∩ Q and v(fD) = −1 if D = D0.

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 SL2(C)-varieties

When H = {Id} : For D ∈ DB, choose fD ∈ C[G] with zero set = D. fD(

• x

y z w

• ) = αx + βy.

The set VG : Determined by values on {fD}, D ∈ DB. v ∈ VG : v(fD0) = r, with r ∈ [−1, 1] ∩ Q and v(fD) = −1 if D = D0. NOTATION v([α : β], r), where fD0 = αx + βy.

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 SL2(C)-varieties

Action of Γ induced by action on B\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 SL2(C)-varieties

Action of Γ induced by action on B\G. σs(([α : β])) = [α : β]. (Fixes P1

R) ;

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 SL2(C)-varieties

Action of Γ induced by action on B\G. σs(([α : β])) = [α : β]. (Fixes P1

R) ;

σs(v([α : β]), r) = v([α : β], r) ;

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 SL2(C)-varieties

Action of Γ induced by action on B\G. σs(([α : β])) = [α : β]. (Fixes P1

R) ;

σs(v([α : β]), r) = v([α : β], r) ; σc(([α : β])) = [β : −α]. (No real fixed points.)

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 SL2(C)-varieties

Action of Γ induced by action on B\G. σs(([α : β])) = [α : β]. (Fixes P1

R) ;

σs(v([α : β]), r) = v([α : β], r) ; σc(([α : β])) = [β : −α]. (No real fixed points.) σc(v([α : β]), r) = v([β : −α].r).

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 SL2(C)-varieties

Action of Γ induced by action on B\G. σs(([α : β])) = [α : β]. (Fixes P1

R) ;

σs(v([α : β]), r) = v([α : β], r) ; σc(([α : β])) = [β : −α]. (No real fixed points.) σc(v([α : β]), r) = v([β : −α].r). For general H : the G-invariant valuations are the restrictions to C(G)H of the G-invariant valuations on C(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 SL2(C)-varieties

An example with H = {Id}

Rn=irreducible G-representation space of dimension n + 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 SL2(C)-varieties

An example with H = {Id}

Rn=irreducible G-representation space of dimension n + 1. X = P(R0 ⊕ R1) × P(R1). G → X with

• a

b c d

• → ([1 : a : c], [b : d]).

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 SL2(C)-varieties

An example with H = {Id}

Rn=irreducible G-representation space of dimension n + 1. X = P(R0 ⊕ R1) × P(R1). G → X with

• a

b c d

• → ([1 : a : c], [b : d]).

Orbits : ℓ1 = {([0 : u1 : u2], [u1 : u2])} ≃ P1 ; ℓ2 = {([1 : 0 : 0], [w1 : w2])} ≃ P1 ; S1 = {([0 : u1 : u2], [w1 : w2])}, with [u1 : u2] = [w1 : w2] S1 ∼ = P1 × P1\ diagonal ; S2 = {([1 : u1 : u2], [u1 : u2])} ∼ = A2 \ {0} . The open orbit U ≃ G, which is the complement of all the

• ther orbits.

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 SL2(C)-varieties

. . . + 1 S1 S2 ℓ2 ℓ1

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 SL2(C)-varieties

. . . + 1 S1 S2 ℓ2 ℓ1 (Any anti-regular involution must stabilize each 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 SL2(C)-varieties

. . . + 1 S1 S2 ℓ2 ℓ1 (Any anti-regular involution must stabilize each orbit.) There exists a (G, σs)-descent datum, with fixed points ∼ = P2(R) × P1(R).

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 SL2(C)-varieties

. . . + 1 S1 S2 ℓ2 ℓ1 (Any anti-regular involution must stabilize each orbit.) There exists a (G, σs)-descent datum, with fixed points ∼ = P2(R) × P1(R). There are no (G, σc)-descent data.

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 SL2(C)-varieties

An example with H = {±Id}

X = P(R1) × P(R1) × P(R1). One open orbit, three 2-dimensional orbits, one 1-dimensional

• rbits :

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 SL2(C)-varieties

An example with H = {±Id}

X = P(R1) × P(R1) × P(R1). One open orbit, three 2-dimensional orbits, one 1-dimensional

• rbits :

Orbits the open orbit ; three orbits isomorphic to P1 × P1\ the diagonal ;

• ne orbit isomorphic to P1.

Any Γ-action permutes the three orbits of dimension 2. Result : X admits exactly two inequivalent (G, σ)-equivariant real structures for each σ ∈ {σs, σc}.

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 SL2(C)-varieties

. . . S2 S3 S1 ℓ

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 SL2(C)-varieties

Case of σ = σs. The open orbit admits two inequivalent (G, σs)

• forms. One does NOT extend to a (G, σs)-form on X. For the
• ther, there are two equivalent (G, σs) forms on the open orbit

which extend to non-equivalent (G, σs)-forms 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 SL2(C)-varieties

Case of σ = σs. The open orbit admits two inequivalent (G, σs)

• forms. One does NOT extend to a (G, σs)-form on X. For the
• ther, there are two equivalent (G, σs) forms on the open orbit

which extend to non-equivalent (G, σs)-forms on X. Real fixed points : the real loci are P1

R × P1 R × P1 R and P1 R × S2,

where S2 is a two-dimensional sphere.

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 SL2(C)-varieties

One can use the action of Γ on the projective line to find all real structures of SL2(C)-almost homogeneous varieties of dimension three.

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 SL2(C)-varieties

Thank you for your attention !

Lucy Moser-Jauslin Forms of almost homogeneous varieties