Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry

Donaldson, Simon; Sun, Song

doi:10.1007/s11511-014-0116-3

Cited by 216 publications

(503 citation statements)

References 30 publications

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“…In this short note we announce a regularity theorem for Kähler-Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of Kähler-Ricci flow on Fano 3-manifolds. Moreover, we also present a partial C 0 estimate to the Kähler-Ricci flow under the regularity assumption, which extends previous works on Kähler-Einstein metrics and shrinking , [11], [20], [15]). The detailed proof will appear in [23].…”

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confidence: 74%

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Regularity of the Kähler–Ricci flow

Тиан

Zhang

2013

Comptes Rendus. Mathématique

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Abstract. In this short note we announce a regularity theorem for Kähler-Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of Kähler-Ricci flow on Fano 3-manifolds. Moreover, we also present a partial C 0 estimate to the Kähler-Ricci flow under the regularity assumption, which extends previous works on Kähler-Einstein metrics and shrinking , [11], [20], [15]). The detailed proof will appear in [23].

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supporting

confidence: 74%

“…The partial C 0 estimate of Kähler-Einstein manifolds plays the key role in Tian's program to resolve the Yau-Tian-Donaldson conjecture, see [17], [18], [19], [11] and [20] for examples. An extension of the partial C 0 estimate to shrinking Kähler-Ricci solitons was given in [15].…”

Section: Regularity Of Kähler-ricci Flowmentioning

confidence: 99%

Regularity of the Kähler–Ricci flow

Тиан

Zhang

2013

Comptes Rendus. Mathématique

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“…Then by [ChC97,ChCT02], we see that J has the C 1,α -regularity on R for every α ∈ (0, 1) (see also [DS14]). In this section we use standard notation in Kähler geometry.…”

Section: Thenmentioning

confidence: 95%

Spectral convergence under bounded Ricci curvature

Honda

2017

Journal of Functional Analysis

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Abstract. For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology by using our scalar curvature. Contents

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“…v is non-zero, but t = 1), generalizing the work of Donaldson-Sun [27]. A modest combination and generalization of these ideas gives the analogous result for the equation (38), and we will give a brief outline of the necessary changes in Section 5.…”

Section: Proof Of the Main Resultsmentioning

confidence: 87%

“…ρ commutes with G. In particular the vector field (F t ) * v along the image F t (M ) is induced by a fixed holomorphic vector field v on P N , since v is G-invariant. We can choose a subsequence t k → T , such that F t k (M ) converges to a limit W ⊂ P N , and as shown in Donaldson-Sun [27], the partial C 0 -estimate implies, up to replacing m by a multiple, that W is a normal Q-Fano variety, homeomorphic to the Gromov-Hausdorff limit Z of the sequence (M, ω t k ). Moreover the maps F t k : M → P N converge to a Lipschitz map F T : Z → P N under this GromovHausdroff convergence, such that F T : Z → W is a homeomorphism.…”

Section: Proof Of the Main Resultsmentioning

confidence: 95%

Kähler–Einstein metrics along the smooth continuity method

2016

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Abstract. We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-DonaldsonSun [17], and can be used to obtain new examples of Kähler-Einstein manifolds. We also give analogous results for twisted Kähler-Einstein metrics and Kahler-Ricci solitons.

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Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry

Cited by 216 publications

References 30 publications

Regularity of the Kähler–Ricci flow

Regularity of the Kähler–Ricci flow

Spectral convergence under bounded Ricci curvature

Kähler–Einstein metrics along the smooth continuity method

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