Stability and coercivity for toric polarizations

Hisamoto, Tomoyuki

doi:10.48550/arxiv.1610.07998

Cited by 28 publications

(60 citation statements)

References 32 publications

(49 reference statements)

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“…Recently there have been significant progresses towards this conjecture, especially on the analytic part (see [6,7,23,24,25,33]) and the Fano case ( [26,67,29]). On the other hand, Berman-Boucksom-Jonsson [4,5,10] proposed a variational approach for attacking this conjecture, which has been successfully carried out in the Fano case, even for singular Fano varieties (see [5,47,50,52] and section 6.1).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Geodesic rays and stability in the cscK problem

Li¹

2020

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We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archimedean entropy functional. As further applications, we show that a uniform K-stability condition for model filtrations and the J K X -stability are both sufficient conditions for the existence of cscK metrics. The first condition is also conjectured to be necessary. Our arguments also produce a different proof of the toric uniform version of YTD conjecture for all polarized toric manifolds. Another result proved here is that the Mabuchi slope of a geodesic ray associated to a test configuration is equal to the non-Archimedean Mabuchi invariant.Conjecture 1.8 (Regularization Conjecture,[20]). For any φ ∈ E 1,NA (L), there exists a sequence {φ m } ⊂ H NA (L) converging to φ in the strong topology such that:Indeed, the results proved here can show that the conjecture 1.8 implies both Conjecture 1.5 and Conjecture 1.6, and hence also the uniform version of YTD conjecture (see Lemma 6.8 and Proposition 6.7). Although Conjecture 1.8 is not known in general yet, we can nevertheless use Boucksom-Jonsson's non-Archimedean approach to K-stability ([19, 20]) and Boucksom-Favre-Jonsson's foundational work on non-Archimedean Monge-Ampère equations ([14, 19, 20]) to get an existence result involving K-stability for model filtrations.To state a general existence statement, let G be a reductive complex Lie group in Aut(X, L) 0 with a maximal compact subgroup K such that K C = G. Let T = ((S 1 ) r ) C be the identity component of the center of

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Geodesic rays and stability in the cscK problem

Li¹

2020

Preprint

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“…Remark 4.2. If A is a constant function, by Theorem 4.1 and the existence of cscK metrics under uniform stability condition for toric manifolds (see [10], [29], [24]), we get the sufficiency of relative K-polystability for the cscK case. By Theorem 4.1, we need to prove the necessity and sufficiency of uniformly relative K-polystability.…”

Section: So We Can Assume Thatmentioning

confidence: 95%

“…By Theorem 4.1, we need to prove the necessity and sufficiency of uniformly relative K-polystability. By Theorem 4.6 in [7] and Lemma 2.8, uniform relative Kpolystability is a necessary condition (also see [24]). In the following, we prove the sufficiency of uniform relative K-polystability.…”

Section: So We Can Assume Thatmentioning

confidence: 97%

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Extremal metrics on toric manifolds and homogeneous toric bundles

Li,

Lian,

Sheng

2021

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“…The question has been settled in some special cases, especially on smooth toric varieties [20,21,33,43,58,69] where the relevant stability notion is expressed in terms of the convex affine geometry of the corresponding Delzant polytope, and is referred to as uniform Kstability of the polytope. Other special cases include Fano manifolds (see e.g.…”

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confidence: 99%

A Yau-Tian-Donaldson correspondence on a class of toric fibrations

Jubert¹

2021

Preprint

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We established a Yau-Tian-Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal Kähler metrics on a large class of toric fibrations, introduced by Apostolov-Calderbank-Gauduchon-Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature Kähler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal Kähler metrics on the total space and a suitable notion of weighted uniform K-stability of the corresponding Delzant polytope. As an application, we show that the projective plane bundle P(L0 ⊕ L1 ⊕ L2), where Li are holomorphic line bundles over an elliptic curve, admits an extremal metric in every Kähler class.

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Stability and coercivity for toric polarizations

Cited by 28 publications

References 32 publications

Geodesic rays and stability in the cscK problem

Geodesic rays and stability in the cscK problem

Extremal metrics on toric manifolds and homogeneous toric bundles

A Yau-Tian-Donaldson correspondence on a class of toric fibrations

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