The Cox ring of a complexity-one horospherical variety

Langlois, Kevin; Terpereau, Ronan

doi:10.1007/s00013-016-0979-y

Cited by 7 publications

(6 citation statements)

References 21 publications

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“…Finally, it would be interesting to combine the ideas from our work with the work of Ilten and Süss [IS] to study varieties with a spherical action of complexity one. In particular, Langlois and Terpereau [LT16,LT] have started a study of horospherical varieties of complexity one that should lead them to a criterion for being Fano, which would be a starting point.…”

Section: Introductionmentioning

confidence: 99%

K-Stability of Fano spherical varieties

Delcroix

2016

Preprint

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We prove a criterion for K-stability of a Q-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds proved by Datar and Székelyhidi, it yields a criterion for the existence of a Kähler-Einstein metric on a spherical Fano manifold. The results hold also for modified K-stability and existence of Kähler-Ricci solitons.The group of G-equivariant automorphisms Aut G (X) of the spherical manifold X is diagonalizable. The real vector space generated by the linear part ofidentified with an element of N − ⊗ R, and a choice ζ ∈ π −1 (ζ) of lift of ζ, we denote by bar DH, ζ (∆ + ) the barycenter of the polytope ∆ + with respect to the measure with density p → e 4ρ−2p, ζ α∈Φ + P κ(α, p) with respect to the Lebesgue measure dp on X(T ) ⊗ R. Our main result is the following.Theorem A. Let X be a Fano spherical manifold. The following are equivalent.(1) There exists a Kähler-Ricci soliton on X with associated complex holomorphic vector field ζ.(2) The barycenter bar DH, ζ (∆ + ) is in the relative interior of the cone 2ρ P + Ξ.(3) The manifold X is modified K-stable with respect to equivariant special test configurations. (4) The manifold X is modified K-stable.The equivalence between (1) and ( 4) holds for any Fano manifold, it is the consequence of the work of Chen, Donaldson and Sun recalled earlier for Kähler-Einstein metrics, and of the work of Datar and Székelyhidi for Kähler-Ricci solitons. When X admits an action of a reductive group G, the equivalence between (1) and ( 3) was shown by Datar and Székelyhidi. What we prove in this article is the equivalence between (2) and (3) in the case of a spherical Fano manifold. Furthermore we prove that the equivalence between (2) and (3) holds for singular Q-Fano spherical varieties.The intuition for our main result came from our previous work on group compactifications, which did not involve K-stability. The proof of a Kähler-Einstein criterion for smooth and Fano group compactifications in [Del15, Del] can be adapted to provide another proof of the criterion for Kähler-Ricci solitons on the same manifolds. Similarly, Wang-Zhu type methods (as used in [WZ04] and [PS10]), together with some results proved for horospherical manifolds in the present paper, could be used to obtain the Kähler-Ricci soliton criterion for these manifolds. One advantage of this other approach is that the value of the greatest Ricci lower bound can be explicitly computed. Alternatively, for horospherical varieties at least, this quantity could be computed using twisted modified K-stability (see [DS]).The computation of the K-stability of a manifold requires two ingredients. The first one is a description of all test configurations (rather, using [DS], all special equivariant test configurations), and the second one is a way to compute the Donaldson-Futaki invariant for all of these test configuration...

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Section: Introductionmentioning

confidence: 99%

K-Stability of Fano spherical varieties

Delcroix

2016

Preprint

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“…Here we focus on the complexity-one torus actions and we freely use the approach of Timashev in [46]. We also refer the reader to [45,37,38,34] for generalizations to other reductive group actions.…”

Section: Preliminariesmentioning

confidence: 99%

“…In relation with mirror symmetry, one may also look at the stringy invariants (see [7]) as studied by Batyrev and Moreau in [8]. Such a description (see [35,36]) is available for complexity-one horospherical varieties [37,38]. Finally, we want to mention that the starting point of the present work comes from the relative version of the h-invariant theory for toric varieties, see [28,15,16].…”

Section: Introductionmentioning

confidence: 99%

Decomposition theorem and torus actions of complexity one

Vicente

Langlois

2020

European Journal of Mathematics

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“…In this case, the Cox ring can be described combinatorially. More generally, horospherical varieties can be described using Cox rings [LT17,Vez20a]. In many cases, computations of intersection theory can be carried out in the Cox ring of an algebraic variety.…”

Section: Introductionmentioning

confidence: 99%

Iteration of Cox rings of klt singularities

Braun,

Moraga

2021

Preprint

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Given a klt singularity (X, ∆; x), we define the iteration of Cox rings of (X, ∆; x). The first result of this article is that the iteration of Cox rings Cox (k) (X, ∆; x) of a klt singularity stabilizes for k large enough. The second result is a boundedness one, we prove that for a n-dimensional klt singularity (X, ∆; x) the iteration of Cox rings stabilizes for k ≥ c(n), where c(n) only depends on n. Then, we use Cox rings to establish the existence of a simply connected factorial canonical cover (or scfc cover) of a klt singularity. We prove that the scfc cover dominates any sequence of quasi-étale finite covers and reductive abelian quasi-torsors of the singularity. We characterize when the iteration of Cox rings is smooth and when the scfc cover is smooth. We also characterize when the spectrum of the iteration coincides with the scfc cover. Finally, we give a complete description of the regional fundamental group, the iteration of Cox rings, and the scfc cover of klt singularities of complexity one. Analogous versions of all our theorems are also proved for Fano type morphisms. To extend the results to this setting, we show that the Jordan property holds for the regional fundamental group of Fano type morphisms.

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The Cox ring of a complexity-one horospherical variety

Cited by 7 publications

References 21 publications

K-Stability of Fano spherical varieties

K-Stability of Fano spherical varieties

Decomposition theorem and torus actions of complexity one

Iteration of Cox rings of klt singularities

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