Quantisation of Extremal Kähler Metrics

Hashimoto, Yoshinori

doi:10.1007/s12220-020-00381-7

Cited by 12 publications

(13 citation statements)

References 52 publications

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“…When we solve the equation (1) for all large enough k in [10], we prove stronger results with more detailed information on the above ξ and c. We consider the following functional.…”

Section: Background On Quantisationmentioning

confidence: 71%

“…Theorem 1.3 is proved in §4, where the definition of relative K-semistability is also provided. In §5 we shall prove Theorem 1.5, and the last section §6 is devoted to the review of the works of [10,22,29,30] from the point of view of relative stability.…”

Section: Organisation Of the Papermentioning

confidence: 99%

“…The main results of [10] that we need can be summarised as follows. Then for all l ∈ N there exists k l ∈ N such that for all k ≥ k l there exists a hermitian matrix H k ∈ B k and A k ∈ θ * (z) such that the following hold:…”

Section: Background On Quantisationmentioning

confidence: 99%

“…When the automorphism group is no longer discrete, a generalisation of Donaldson's result was established in [10], widening the scope to include extremal Kähler metrics. This was done by considering an equation…”

Section: Introductionmentioning

confidence: 95%

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Relative stability associated to quantised extremal Kähler metrics

Hashimoto¹

2019

J. Math. Soc. Japan

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We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and K-semistability of extremal manifolds by using quantised extremal metrics; this gives an alternative proof of the results of Mabuchi and Stoppa-Székelyhidi. In proving them, we further provide an explicit local density formula for the equivariant Riemann-Roch theorem.Then (X, L k ) is weakly Chow polystable relative to the centre of K.We shall see in the proof that the converse does not hold in general; the solvability of (1) is strictly stronger than weak relative Chow polystability (cf. Remark 3.11).The main result of [10] is that (1) is solvable for all large enough k, if (X, L) admits an extremal metric (cf. Theorem 2.9). Combining the main result of [10] and Theorem 1.1, we obtain the following corollary.Corollary 1.2. If a polarised Kähler manifold (X, L) admits an extremal Kähler metric, then it is asymptotically weakly Chow polystable relative to the centre of K.This corollary is also a consequence of the works of Mabuchi [17,18,19]. A stronger version of the above corollary was recently proved by Mabuchi [22] (see also [30]).Our second application to stability is the following.

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“…When we solve the equation (1) for all large enough k in [10], we prove stronger results with more detailed information on the above ξ and c. We consider the following functional.…”

Section: Background On Quantisationmentioning

confidence: 71%

Section: Organisation Of the Papermentioning

confidence: 99%

Section: Background On Quantisationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

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Relative stability associated to quantised extremal Kähler metrics

Hashimoto¹

2019

J. Math. Soc. Japan

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“…The proof also relied on the balancing flow method, which was further exploited in several settings (e.g. [Fin10,Has15]). We remark that the notion of Donaldson's balanced metric does not agree with that introduced by the author.…”

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confidence: 99%

Ricci Iteration for Coupled Kähler–Einstein Metrics

Takahashi

2019

International Mathematics Research Notices

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We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the "balanced metrics". We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern class, we introduce some algebro-geometric obstruction which interpolates between the Donaldson-Futaki invariant and Chow weight. Then we show the existence and weak convergence of balanced metrics on CKE manifolds under the vanishing of this obstruction. Moreover, restricted to the case when the automorphism group is discrete, we also discuss approximate solutions and a gradient flow method towards the smooth convergence.

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Balanced Metrics for Extremal Kähler Metrics and Fano Manifolds

Hashimoto

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Quantisation of Extremal Kähler Metrics

Cited by 12 publications

References 52 publications

Relative stability associated to quantised extremal Kähler metrics

Relative stability associated to quantised extremal Kähler metrics

Ricci Iteration for Coupled Kähler–Einstein Metrics

Balanced Metrics for Extremal Kähler Metrics and Fano Manifolds

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