Canonical measures and Kähler-Ricci flow

Song, Jian; Тиан, Ганг

doi:10.1090/s0894-0347-2011-00717-0

Cited by 193 publications

(274 citation statements)

References 39 publications

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“…[5,6,9]) to list some problems and speculations. Some of these problems are doable by adapting arguments from what has been done for the Kähler-Ricci flow (cf.…”

Section: Lemma 32 Let E Be a Divisor In A Projective Manifold M If mentioning

confidence: 99%

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A continuity method to construct canonical metrics

Nave

Тиан

2015

Math. Ann.

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“…[5,6,9]) to list some problems and speculations. Some of these problems are doable by adapting arguments from what has been done for the Kähler-Ricci flow (cf.…”

Section: Lemma 32 Let E Be a Divisor In A Projective Manifold M If mentioning

confidence: 99%

“…There has been quite a bit of progresses and many very important results have been proven (e.g. [5,6,10]). In this short paper, we introduce a new continuity method which provides an alternative way of carrying out the Analytic Minimal Model G. Tian was supported partially by grants from NSF and NSFC.…”

Section: Introductionmentioning

confidence: 99%

A continuity method to construct canonical metrics

Nave

Тиан

2015

Math. Ann.

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“…Let α 0 be an ample class on N , and ω t ∈ f * α 0 + tα be the Ricci-flat Kähler metric given by Yau's Theorem [40] for t ∈ (0, 1], which satisfies the complex Monge-Ampère equation 2 Ω ∧ Ω. By [35] and [12],ω t converges smoothly to f * ω on f −1 (K) for any compact K ⊂ N 0 as t → 0, and on N 0 , where ω is the Kähler metric on N 0 with Ric(ω) = ω W P obtained in [35] and [29] (see also [28]), and ω W P is a Weil-Petersson semipositive form on N 0 coming from the variation of the complex structures of the fibers M y . Furthermore, the Ricci-flat metricsω t have locally uniformly bounded curvature on f −1 (N 0 ).…”

Section: Introductionmentioning

confidence: 97%

Gromov–Hausdorff collapsing of Calabi–Yau manifolds

Gross

Tosatti

Zhang

2016

Communications in Analysis and Geometry

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Abstract. This paper is a sequel to [12]. We further study GromovHausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as [12], if the dimension of the base manifold is one, the limit metric space is homeomorphic to the base manifold. Secondly, if the fibered Calabi-Yau manifolds are Lagrangian fibrations of holomorphic symplectic manifolds, the metrics on the regular parts of the limits are special Kähler metrics. By combining these two results, we extend [13] to any fibered projective K3 surface without any assumption on the type of singular fibers.

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“…Song and G. Tian [ST09], [ST12]. It requires to study the behavior of the Kähler-Ricci flow on mildly singular varieties, and one is naturally lead to study weak solutions of degenerate complex Monge-Ampère flows (when the function F in (CM AF ) is not smooth but continuous).…”

Section: Introductionmentioning

confidence: 99%

Regularizing properties of complex Monge–Ampère flows

Tô

2017

Journal of Functional Analysis

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Abstract. We study the regularizing properties of complex MongeAmpère flows on a Kähler manifold (X, ω) when the initial data are ω-psh functions with zero Lelong number at all points. We prove that the general Monge-Ampère flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solution. IntroductionLet (X, ω) be a compact Kähler manifold of complex dimension n and α ∈ H 1,1 (X, R) a Kähler class with ω ∈ α. Let Ω be a smooth volume form on X. Denote by (θ t ) t∈[0,T ] a family of Kähler forms on X, and assume that θ 0 = ω. The goal of this note is to prove the regularizing and stability properties of solutions to the following complex Monge-Ampère flowwhere F is a smooth function and ϕ(0, z) = ϕ 0 (z) is a ω-plurisubharmonic (ω-psh) function with zero Lelong numbers at all points.One motivation for studying this Monge-Ampère flow is that the Käler-Ricci flow can be reduced to a particular case of (CM AF ). When F = F (z) and θ t = ω + tχ, where χ = η − Ric(ω), then (CM AF ) is the local potential equation of the twisted Kähler-Ricci flow

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Canonical measures and Kähler-Ricci flow

Cited by 193 publications

References 39 publications

A continuity method to construct canonical metrics

A continuity method to construct canonical metrics

Gromov–Hausdorff collapsing of Calabi–Yau manifolds

Regularizing properties of complex Monge–Ampère flows

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