On the Yau-Tian-Donaldson conjecture for singular Fano varieties

Li, Chi; Тиан, Ганг; Wang, Feng

doi:10.48550/arxiv.1711.09530

Cited by 18 publications

(29 citation statements)

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“…In fact we expect that our approach can be generalized to the case of big line bundles, yielding new existence results for the general Monge-Ampère equations considered in [11], and answering some questions proposed in [42,Section 6.3]. Another direction to pursue would be to consider the the case of singular varieties (as in [36,31,32]) or the equivariant case (as in [29,27]).…”

Section: Resultsmentioning

confidence: 71%

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A quantization proof of the uniform Yau-Tian-Donaldson conjecture

Zhang¹

2021

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Using quantization techniques, we show that the δ-invariant of Fujita-Odaka coincides with the optimal exponent in certain Moser-Trudinger type inequality. Consequently we obtain a uniform Yau-Tian-Donaldson theorem for the existence of twisted Kähler-Einstein metrics with arbitrary polarizations. Our approach mainly uses pluripotential theory, which does not involve Cheeger-Colding-Tian theory or the non-Archimedean language. A new computable criterion for the existence of constant scalar curvature Kähler metrics is also given.

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Section: Resultsmentioning

confidence: 71%

“…By using continuity methods (cf. [40,16,19,31,41]) or the variational approach (cf. [2,32,29]), we now have a fairly good understanding of the YTD conjecture in this scenario.…”

Section: Setup and The Main Resultsmentioning

confidence: 99%

A quantization proof of the uniform Yau-Tian-Donaldson conjecture

Zhang¹

2021

Preprint

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“…In this paper, we are interested in the generalized Yau-Tian-Donaldson conjecture meaning that X is allowed to be singular. There are some previous works [40,38] and [37] on extending the YTD conjecture to special classes of singular Fano varieties. Berman's work in [1] shows that the "only if" part of the conjecture is indeed true for any log Fano pair.…”

Section: Introductionmentioning

confidence: 99%

“…It's well known that to solve Kähler-Einstein metrics on singular varieties is equivalent to solve some degenerate Monge-Ampère equation on a resolution of the variety (see [29,4]). It's natural to study such degenerate Monge-Amère equation using an appropriate sequence of non-degenerate Monge-Ampère equations to approximate the original equation, which is the guiding principle in [37]. The perturbative approach used here is motivated by this idea but is more on the non-Archimedean side.…”

Section: Introductionmentioning

confidence: 99%

“…In the next section, we will first discuss Berman-Boucksom-Jonsson's variational approach to YTD and our previous work in [37] which uses perturbation arguments to prove YTD for a class of singular Fano varieties. These will serve as comparisons to our new argument to get the uniform version of YTD in the singular case, which we sketch in section 2.3 highlighting some new ingredients about convergence of slope and non-Archimedean quantities.…”

Section: Introductionmentioning

confidence: 99%

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The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties

Тиан

Wang

2019

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Delta Invariants of Singular del Pezzo Surfaces

2020

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We use the methods introduced by Cheltsov-Rubinstein-Zhang in [CRZ18] to estimate δ-invariants of the seven singular del Pezzo surfaces with quotient singularities studied by Cheltsov-Park-Shramov in [CPS10] that have α-invariants less than 2 3 . As a result, we verify that each of these surfaces admits an orbifold Kähler-Einstein metric.All varieties are assumed to be complex, projective and normal unless otherwise stated.2010 Mathematics Subject Classification. 14J17, 14J45, 32Q20. 1 holds, where α(S d ) is the α-invariant of the surface S d . Indeed, Johnson and Kollár verified (1.3) in the case when I = 1, the surface S d is singular, and the quintuple (a 0 , a 1 , a 2 , a 3 , d) is not one of the four exceptions (1.2). Two of the four remaining cases (1.2) have been treated in [A02] by Araujo, who proved the following result:Theorem 1.4 ([A02, Theorem 4.1]). In the following two cases:• (a 0 , a 1 , a 2 , a 3 , d) = (1, 2, 3, 5, 10),• (a 0 , a 1 , a 2 , a 3 , d) = (1, 3, 5, 7, 15) and the equation of S d contains yzt, the inequality α(S d ) > 2 3 holds. In particular, S d admits an orbifold Kähler-Einstein metric. The remaining two cases of (1.2) have been dealt with in the paper [CPS10], which was published in Journal of Geometric Analysis in 2010. In this paper, we succeeded in estimating their α-invariants from below by large enough numbers for the criterion (1.3). To be precise, we proved the following result:Theorem 1.5 ([CPS10, Theorem 1.10]). Suppose that (a 0 , a 1 , a 2 , a 3 , d) = (1, 3, 5, 8, 16) or (2, 3, 5, 9, 18). Then α(S d ) > 2 3 . In particular, S d admits an orbifold Kähler-Einstein metric. In particular, if I = 1, then S d admits an orbifold Kähler-Einstein metric except possibly the case when (a 0 , a 1 , a 2 , a 3 , d) = (1, 3, 5, 7, 15) and the defining equation of the surface S d does not contain yzt. Note that in the latter case one has α(S d ) = 8 15 < 2 3 by [CPS10, Theorem 1.10], so that the criterion by the α-invariant could not be applied.Meanwhile, since 2010 we have witnessed dramatic developments in the study of the Yau-Tian-Donaldson conjecture concerning the existence of Kähler-Einstein metrics on Fano manifolds and stability. The challenge to the conjecture has been heightened by Chen, Donaldson, Sun and Tian who have completed the proof for the case of Fano manifolds with anticanonical polarisations [CDS15, T15]. Following this celebrated achievement, useful technologies have been introduced to determine whether given Fano varieties are Kähler-Einstein or not, via the theorem of Chen-Donaldson-Sun and Tian.Recently Fujita and Odaka introduced a new invariant of Fano varieties, which they called δ-invariant (for the definition, see [FO18, Definition 1.2]), that serves as a strong criterion for uniform K-stability (see [FO18]). Theorem 1.6 ([FO18, BJ17]). Let X be a Fano variety with at most Kawamata log terminal singularities. Then X is uniformly K-stable if and only if δ(X) > 1. This powerful tool has been practiced for del Pezzo surfaces in [PW18, CRZ18, CZ18]. Around...

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On the Yau-Tian-Donaldson conjecture for singular Fano varieties

Cited by 18 publications

References 53 publications

A quantization proof of the uniform Yau-Tian-Donaldson conjecture

A quantization proof of the uniform Yau-Tian-Donaldson conjecture

The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties

Delta Invariants of Singular del Pezzo Surfaces

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