Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds

Delcroix, Thibaut; Hultgren, Jakob

doi:10.1016/j.matpur.2020.12.002

Cited by 12 publications

(21 citation statements)

References 24 publications

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“…The advantage here is that no holomorphic vector field is involved, and the drawback is that several Kähler classes have to be considered. Until the present paper, only two examples of Fano manifolds with no Kähler-Einstein metrics but a pair of coupled Kähler-Einstein metrics were known [10,8].…”

Section: Date: 2019mentioning

confidence: 91%

“…In the body of the paper we will prove the results stated in this introduction. The results to prove are translated into combinatorial conditions by using [8] or earlier results. As already mentioned, it does not mean that checking the conditions is trivial, as soon as we consider infinite families.…”

Section: Date: 2019mentioning

confidence: 99%

“…As a consequence, H contains the unipotent radical of P , and G/H is a rank one horospherical homogeneous space. To be consistent with the conventions in [8] and [7], we fix an involution σ of t such that t σ = ker(α 1,p+1 ). It amounts to choosing the fixed point set of σ, a complement to Rα 1,p+1 in X(T ) ⊗ R. Here, we choose the orthogonal of α 1,p+1 with respect to the Killing form to define σ.…”

Section: 21mentioning

confidence: 99%

“…to ensure again consistency with the conventions in [8] and [7]. The spherical lattice M is then X(T /T ∩ H) = Zα 1,p+1 .…”

Section: 21mentioning

confidence: 99%

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Examples of K-unstable Fano manifolds

Delcroix¹

2019

Preprint

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We examine various examples of horosymmetric manifolds which exhibit interesting properties with respect to canonical metrics. In particular, we determine when the blow-up of a quadric along a linear subquadric admits Kähler-Einstein metrics, providing infinitely many examples of manifolds with no Kähler-Ricci solitons that are not Ksemistable. Using a different construction, we provide an infinite family of Fano manifolds with no Kähler-Einstein metrics but which admit coupled Kähler-Einstein metrics. Finally, we elaborate on the relationship between Kähler-Ricci solitons and the more general concept of multiplier Hermitian structures and illustrate this with examples related to the two previous families.

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Section: Date: 2019mentioning

confidence: 91%

Section: Date: 2019mentioning

confidence: 99%

Section: 21mentioning

confidence: 99%

“…to ensure again consistency with the conventions in [8] and [7]. The spherical lattice M is then X(T /T ∩ H) = Zα 1,p+1 .…”

Section: 21mentioning

confidence: 99%

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Examples of K-unstable Fano manifolds

Delcroix¹

2019

Preprint

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“…When X is a smooth Fano equivariant compactifications of complex Lie groups, Delcroix [Del17] obtained a formula for the greatest Ricci lower bound, where the barycenter of the moment polytope with respect to the Lebesgue measure is replaced by the barycenter with respect to the Duistermaat-Heckman measure. Furthermore, the recent result of Delcroix and Hultgren [DH20] extend the formula to the case of horosymmetric manifolds introduced in [Del20b], which is a class of spherical varieties including horospherical manifolds and smooth projective symmetric varieties. Since any Q-Fano spherical variety admits a special test configuration with horospherical central fiber by [Del20a,Corollary 3.31], for the Donaldson-Futaki invariants of Q-Fano spherical varieties, it is enough to calculate them in the case of horospherical varieties.…”

Section: Introductionmentioning

confidence: 95%

Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one

Hwang¹,

Kim²,

Park³

2021

Preprint

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A horospherical variety is a normal G-variety such that a connected reductive algebraic group G acts with an open orbit isomorphic to a torus bundle over a rational homogeneous manifold. The projective horospherical manifolds of Picard number one are classified by Pasquier, and it turned out that the automorphism groups of all nonhomogeneous ones are non-reductive, which implies that they admit no Kähler-Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be Kähler-Einstein, we compute the greatest Ricci lower bounds of projective horospherical manifolds of Picard number one using the barycenter of each moment polytope with respect to the Duistermaat-Heckman measure based on a recent work of Delcroix and Hultgren. In particular, the greatest Ricci lower bound of the odd symplectic Grassmannian SGr(n, 2n + 1) can be arbitrarily close to zero as n grows.

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Calabi type functionals for coupled Kähler–Einstein metrics

Nakamura

2023

Ann Glob Anal Geom

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We introduce the coupled Ricci–Calabi functional and the coupled H-functional which measure how far a Kähler metric is from a coupled Kähler–Einstein metric in the sense of Hultgren–Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give corresponding Hessian formulas for these functionals at each critical point, which have an application to a Matsushima type obstruction theorem for the existence of a coupled Kähler–Einstein metric.

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Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds

Cited by 12 publications

References 24 publications

Examples of K-unstable Fano manifolds

Examples of K-unstable Fano manifolds

Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one

Calabi type functionals for coupled Kähler–Einstein metrics

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