Combinatorial characterization of the weight monoids of smooth affine spherical varieties

Pezzini, Guido; Steirteghem, Bart Van

doi:10.1090/tran/7785

Cited by 10 publications

(18 citation statements)

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“…In this section we begin by briefly reviewing notions from the combinatorial theory of affine spherical varieties we will need. For further details and context, we refer to [BVS16] and [PVS16]. In Theorem 2.18, we recall from [PVS16] the characterization of G-saturated weight monoids of smooth affine spherical varieties.…”

Section: Smooth Affine Spherical Varietiesmentioning

confidence: 99%

“…For further details and context, we refer to [BVS16] and [PVS16]. In Theorem 2.18, we recall from [PVS16] the characterization of G-saturated weight monoids of smooth affine spherical varieties. This will be the main tool we will use in Section 3.…”

Section: Smooth Affine Spherical Varietiesmentioning

confidence: 99%

“…For any given pair (P, Λ 0 ) and vertex a of P, checking the conditions (1.4) can be reduced to a finite number of elementary verifications. Indeed, in [PVS16] Pezzini and Van Steirteghem used the combinatorial theory of spherical varieties (always defined over C in this paper) and a smoothness criterion due to R. Camus [Cam01] to give a combinatorial characterization of the weight monoids of smooth affine spherical varieties.…”

Section: Introductionmentioning

confidence: 99%

“…Put differently, the classification of smooth affine spherical varieties is equivalent to the classification of their weight monoids. In this paper we apply the criterion in [PVS16] to classify the weight monoids of three families of smooth affine spherical varieties.…”

Section: Introductionmentioning

confidence: 99%

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On Some Families of Smooth Affine Spherical Varieties of Full Rank

Paulus

Pezzini

Steirteghem

2018

Acta. Math. Sin.-English Ser.

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Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify: (a) all such varieties for G = SL(2) × C × and (b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and F. Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of C. Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.Correspondingly, the varieties X a can be considered as "building blocks" for the manifold M. Let us recall briefly in which sense, referring to Section 2 of [Kno11] for details: the manifold M admits an open cover M = a U a , where the union is over all vertices a of P M and U a is K-stable for all a. Moreover, the subset U a is a K-equivariant fibre bundle of the form K × Ka Y a , where Y a is a K a -stable open subset of X a intersecting the closed G(a)-orbit.Example 1.1. Let K = SU(2). Then one can identify su(2) * with R 3 and the coadjoint action of SU(2) with the standard action through SUHamiltonian manifold under the diagonal action of SU(2) with moment mapOne can check that M is multiplicity free. For T R we can take the subgroup of K made up of diagonal matrices. Then T R ∼ = U(1) and Λ ∼ = Z. We can choose the identification su(2) * ≡ R 3 in such a way that Lie(T R ) * gets identified with R(0, 0, 1) and the weight lattice Λ with Z(0, 0, 1). As a dominant Weyl chamber t + we can then choose R ≥0 (0, 0, 1). To lighten notation, we will identify Λ with Z and t + with R ≥0 . Elementary computations show thatwhere b ∈ R >0 depends on the choice of v ∈ su(2) * , that L R = T R and that the generic isotropy group K M is equal to the center of SU(2). Consequently,We confirm that the conditions (1.4) are satisfied at the two vertices of P M . We begin with the vertex 0. Since K 0 = SU(2), we have that G(0) = SL(2). Observe that C 0 P = t + . Consider the smooth affine spherical SL(2)-varietywhere T is the subset of SL(2) made up of diagonal matrices. As is well known, the weight monoid of X 0 is 2N ⊂ N = Λ ∩ t + , whence (1.4) holds at the vertex 0. Next we consider the vertex b of P M . Then K b = T R and consequently G(b) = T . Note that C b P M = −t + . Straightforward computations show that for the smooth affine spherical T -varietythe conditions in (1.4) hold again. Remark 1.2. (a) Generalizing Example 1.1, Corollary 11.4 in [Kno11] shows how one can recover P. Iglésias's classification of multiplicity free Hamiltonian SU(2)-manifolds in [Igl91] from Knop's theorems above. (b) Similarly, Delzant's classification of symplectic toric manifolds in [Del88] by Delzant polytopes is an immediate consequence of Knop's results (see [Kno11, Corollary 11.3].) (c) In [Kno16, Section ...

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Section: Smooth Affine Spherical Varietiesmentioning

confidence: 99%

Section: Smooth Affine Spherical Varietiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Some Families of Smooth Affine Spherical Varieties of Full Rank

Paulus

Pezzini

Steirteghem

2018

Acta. Math. Sin.-English Ser.

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“…On the other hand, Theorem 3.22], we see that soc(v 2 ) is not the socle of a spherical module. Indeed, the only socle in Table 2 of [PVS19] which has the same first four entries as soc(v 2 ) (up to isomorphism of socles) is socle #6. On the other hand, because ρ(D + 2 ), α 1 + α 3 = −3 = −1, the socle soc(v 2 ) is not isomorphic to socle #6 of [PVS19, Table 2].…”

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Momentum polytopes of projective spherical varieties and related Kähler geometry

Cupit-Foutou

Pezzini

Steirteghem

2020

Sel. Math. New Ser.

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We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, as well as a classification of all Fano spherical varieties. In the setting of multiplicity free compact and connected Hamiltonian manifolds, we obtain a necessary and sufficient condition involving momentum polytopes for such manifolds to be Kähler and classify the invariant compatible complex structures of a given Kähler multiplicity free compact and connected Hamiltonian manifold.2010 Mathematics Subject Classification. Primary 14M27, 53D20, 32Q15. Key words and phrases. Spherical variety, momentum polytope, multiplicity free Hamiltonian manifold, multiplicity free Kähler manifold. 1As proved by A. Huckleberry and T. Wurzbacher in [HW90], Kähler multiplicity free Kmanifolds are spherical K C -varieties (i.e. contain a dense orbit of a Borel subgroup of K C ). Using this fact and applying the Uniqueness Theorem on smooth affine spherical varieties he obtained in [Los09a], I. Losev proved in loc.cit. a refined uniqueness result for invariant Kähler structures on a given compact multiplicity free Hamiltonian K-manifold.One of the results in this paper is a combinatorial answer to the Kählerizability question, which was first studied in [Woo98b]. Considering Brion's result [Bri87] asserting that a smooth projective K C -variety is multiplicity free as a K-Hamiltonian manifold if and only if it is spherical as a K C -variety, we approach this question by making use of the theory of projective spherical varieties, just like Woodward did in [Woo98b]. In fact, most of the current paper's results are about these varieties. In particular, we obtain a combinatorial classification of the (not necessarily smooth) polarized spherical varieties involving rational convex polytopes. This classification parallels the well known classification of polarized toric varieties in terms of integral convex polytopes. Moreover, it enables us to derive a classification of Kähler multiplicity free compact and connected Hamiltonian manifolds, which generalizes Delzant's classification of toric manifolds.One important ingredient in this work is the momentum polytope Q(X, L) of a polarized K C -variety (X, L) introduced by M. Brion in [Bri87]. This polytope turns out to be a purely algebraic version of the Kirwan polytope. In Section 2, we recall Brion's representationtheoretic definition of a momentum polytope and its main properties. In Section 3, we focus on polarized spherical varieties; we collect some known facts on the combinatorial invariants of these varieties and their intertwining relations with momentum polytopes.Section 4 is the heart of this paper. Here we use the combinatorial theory of spherical varieties, including their recent complete combinatorial classification to characterize the momentum polytopes of polarized spherical varieties. More specifically, we give two combinatorial characterizations of the pairs (Ξ, Q), where...

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Equivariant Models of Spherical Varieties

Borovoi

2019

Transformation Groups

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Let k0 be a field of characteristic 0 with algebraic closure k. Let G be a connected reductive k-group, and let Y be a spherical variety over k (a spherical homogeneous space or a spherical embedding). Let G0 be a k0-model (k0-form) of G. We give necessary and sufficient conditions for the existence of a G0-equivariant k0-model of Y .Let Y be an algebraic variety over k.where Y 0 is an algebraic variety over k 0 andis an isomorphism of k-varieties. By abuse of language, we say just thatNote that for any such isomorphism β 0 : Y 0 ∼ − → Y 0 there exists a unique automorphism β : Y ∼ − → Y such that the following diagram commutes:It is a classical problem of Algebraic Geometry and Galois Cohomology to describe the set of isomorphism classes of k 0 -models of a k-variety Y . If there exists one such model (Y 0 , ν Y ), then it defines an injective map from the set of all isomorphism classes of k 0 -models into the first Galois cohomology set H 1 (k 0 , Aut(Y 0 )). Here we denote by Aut(Y ) the group of automorphisms of Y (regarded as an abstract group) and we write Aut(Y 0 ) for the group Aut(Y ) equipped with the Galois action induced by the k 0.3.When Y carries an additional structure, we wish Y 0 to carry this structure as well. For example, if G is an algebraic group over k, we define a k 0 -model of G as a pairis an isomorphism of algebraic k-groups. We define isomorphisms of k 0 -models of G similarly to the definition in Subsection 0.1. If there exists a k 0 -model (G 0 , ν G ), then it defines a bijection between the set of isomorphism classes of k 0 -models of G and the first Galois cohomology set H 1 (k 0 , Aut(G 0 )); see Serre [Ser97, III.1.3, Corollary of Proposition 5]. 0.4. Let G be a (connected) reductive group over k and Y is a k-variety equipped with a G-action θ : G × k Y → Y . This means that θ is a morphism of k-varieties such that certain natural diagrams commute; see Milne [Mil17, Section 1.f]. We say then that Y is a G-k-variety. By a k 0 -model of (G, Y, θ) we mean a triple (G 0 , Y 0 , θ 0 ) as above, but over k 0 , together with two isomorphismsν G is an isomorphism of algebraic k-groups and ν Y is an isomorphism of k-varieties, such that the following diagram commutes:We define isomorphisms of k 0 -models of (G, Y, θ) in a natural way. We wish to classify isomorphism classes of k 0 -models (G 0 , Y 0 , θ 0 ) of triples (G, Y, θ).It is well known (see, for instance, Milne [Mil17, Theorem 23.55]) that G admits a split k 0 -model G spl (also called the Chevalley form). Then one can classify k 0 -models of G by the cohomology classes in H 1 (k 0 , Aut(G spl )). One can also classify k 0 -models k U /k 0 with finite Galois group G/U, which is classical; see, for instance, Jahnel [Jah, Proposition 2.3]. 7.11. Given any k 0 -point of (M Γ ) 0 of the form [X, τ ], it will be straightforward to obtain a G 0 -equivariant k 0 -model of X (this will be done in the proof of Theorem 7.18). In order to prove that such a k 0 -point exists, our plan is to consider a certain subscheme C X of M Γ , which wi...

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Combinatorial characterization of the weight monoids of smooth affine spherical varieties

Cited by 10 publications

References 33 publications

On Some Families of Smooth Affine Spherical Varieties of Full Rank

On Some Families of Smooth Affine Spherical Varieties of Full Rank

Momentum polytopes of projective spherical varieties and related Kähler geometry

Equivariant Models of Spherical Varieties

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