Finite generation for valuations computing stability thresholds and applications to K-stability

Liu, Yuchen; Xu, Chenyang; Zhuang, Ziquan

doi:10.48550/arxiv.2102.09405

Cited by 19 publications

(36 citation statements)

References 27 publications

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“…There is a folklore conjecture that the uniform K-stability is equivalent to the K-stability for log pairs. There is a partial consequence: a log Fano pair (X, B, −K (X,B) ) is K-stable if and only if it is uniformly K-stable due to [33] (for smooth Fano manifolds, cf. [8], [47] and [3]).…”

Section: Introductionmentioning

confidence: 99%

A decomposition formula for J-stability and its applications

Hattori¹

2021

Preprint

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For algebro-geometric study of J-stability, a variant of K-stability, we prove a decomposition formula of non-archimedean J -energy of n-dimensional varieties into n-dimensional intersection numbers rather than (n + 1)-dimensional ones, and show the equivalence of slope J H -(semi)stability and J H -(semi)stability for surfaces when H is pseudoeffective. Among other applications, we also give a purely algebro-geometric proof of a uniform K-stability of minimal surfaces due to [23], and provides examples which are J-stable (resp., K-stable) but not uniformly J-stable (resp., uniformly K-stable).

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Section: Introductionmentioning

confidence: 99%

A decomposition formula for J-stability and its applications

Hattori¹

2021

Preprint

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“…An almost complete theory of K-stability of Fano varieties is established. From this powerful theory, one can construct a satisfactory moduli space of K-stable Fano varieties, the so-called K-moduli space which is proper as proved recently in [32]. There are many works along these lines, see [7], [15], [6], [41], [1], [4], [42], etc.…”

Section: Introductionmentioning

confidence: 98%

Openness of uniformly valuative stability on the Kähler cone of projective manifolds

Liu¹

2021

Preprint

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Assume that a projective variety with a polarization is uniformly valuatively stable (stronger than uniformly valuatively stable given by Dervan and Legendre in [19]), then the projective variety with any polarization sufficiently close to the original polarization is uniformly valuatively stable. We also define valuative stability for the transcendental Kähler classes. Our openness result can be extended to the Kähler cone of projective manifolds.

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“…According to the Yau-Tian-Donaldson conjecture a Fano variety X admits a Kähler-Einstein metric if and only if (X, −K X ) is K-polystable. The cases when X is non-singular was established in [14] and, very recently, the singular case was settled in [23]. As for the notion of Gibbs stability it arose in the probabilistic construction of Kähler-Einstein metrics on Fano manifolds proposed in [5].…”

Section: Introductionmentioning

confidence: 99%

Measure preserving holomorphic vector fields, invariant anti-canonical divisors and Gibbs stability

Berman¹

2022

Preprint

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Let X be a compact complex manifold whose anti-canonical line bundle −KX is big. We show that X admits no non-trivial holomorphic vector fields if it is Gibbs stable (at any level). The proof is based on a vanishing result for measure preserving holomorphic vector fields on X of independent interest. As an application it shown that, in general, if −KX is big, there are no holomorphic vector fields on X that are tangent to a non-singular irreducible anti-canonical divisor S on X. More generally, the result holds for varieties with log terminal singularities and log pairs. Relations to a result of Berndtsson about generalized Hamiltonians and coercivity of the quantized Ding functional are also pointed out.

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Finite generation for valuations computing stability thresholds and applications to K-stability

Cited by 19 publications

References 27 publications

A decomposition formula for J-stability and its applications

A decomposition formula for J-stability and its applications

Openness of uniformly valuative stability on the Kähler cone of projective manifolds

Measure preserving holomorphic vector fields, invariant anti-canonical divisors and Gibbs stability

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