Kähler-Ricci flow on projective bundles over Kähler-Einstein manifolds

Fong, Frederick

doi:10.1090/s0002-9947-2013-05726-1

Cited by 11 publications

(9 citation statements)

References 19 publications

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“…Since χ Y is rational and 0 < C −1 ≤ (a) Case 0 < k < n. If additionally Y is smooth and f : X → Y is a submersion, then χ Y is a Kähler metric on Y and ω(t) → f * χ Y in C 0 (X, ω 0 )-topology exponentially fast (see Theorem 1.2 (2.2)) and hence (X, ω(t)) → (Y, χ Y ) in Gromov-Hausdorff topology. This provides an alternative way (using the twisted Kähler-Ricci flow) to deform a Fano bundle to the base in Gromov-Hausdorff topology (compare with the works in [8,10,26]). (b) Case k = n. In this case χ Y is in fact solved by (f * χ Y + √ −1∂∂ψ) n = e ψ Ω on X.…”

Section: Remarks On the Twisted Kähler-ricci Flowmentioning

confidence: 99%

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On a twisted conical Kähler–Ricci flow

Zhang

2018

Ann Glob Anal Geom

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In this paper, we discuss diameter bound and Gromov-Hausdorff convergence of a twisted conical Kähler-Ricci flow on the total spaces of some holomorphic submersions. We also observe that, starting from a model conical Kähler metric with possibly unbounded scalar curvature, the conical Kähler-Ricci flow will instantly have bounded scalar curvature for t > 0, and the bound is of the form C t . Several key results will be obtained by direct arguments on the conical equation without passing to a smooth approximation. In the last section, we present several remarks on a twisted Kähler-Ricci flow and its convergence.

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Section: Remarks On the Twisted Kähler-ricci Flowmentioning

confidence: 99%

“…There are several progresses on above item (2), see e.g. [8,10,26,28,37]. However, it seems in general the finite time convergence of the Kähler-Ricci flow are still unclear.…”

Section: Remarks On the Twisted Kähler-ricci Flowmentioning

confidence: 99%

On a twisted conical Kähler–Ricci flow

Zhang

2018

Ann Glob Anal Geom

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“…It was shown by Song-Székelyhidi-Weinkove [59] that the diameter of the collapsing fiber is bounded above by a multiple of (T − t) 1/3 , but this falls short of the optimal rate of (T − t) 1/2 (see also [21]). It is expected that the blow-up limit is a product with a flat direction and this has been proved under symmetry conditions [20,57]. Underlying this difficulty is the depth of the problem of understanding the Kähler-Ricci flow for a general initial metric on P 1 (the result of Hamilton and Chow), which still has no simple proof.…”

Section: Some Open Problemsmentioning

confidence: 99%

The Kähler-Ricci flow on compact Kähler manifolds

Weinkove¹

2016

Geometric Analysis

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These lecture notes are based on five hours of lectures given at the Park City Math Institute in the summer of 2013. The notes are intended to be a leisurely introduction to the Kähler-Ricci flow on compact Kähler manifolds. They are aimed at graduate students who have some background in differential geometry, but do not necessarily have any knowledge of Kähler geometry or the Ricci flow. There are exercises throughout the text. The goal is that by the end, the reader will learn the basic techniques in the Kähler-Ricci flow and know enough to be able to explore the current literature.The material covered by these notes is as follows. In the first lecture, we give a quick introduction to some of the main definitions and tools of Kähler geometry. In Lecture 2, we introduce the Kähler-Ricci flow and give some simple examples, before stating, and in Lecture 3 proving, the maximal existence time theorem for the flow. In Lecture 4 we prove long time convergence results in the cases when the manifold has negative or zero first Chern class. Finally in Lecture 5 we discuss more recent work on the behavior of the flow on Kähler surfaces. We also "go beyond" the Kähler-Ricci flow by discussing a new flow on complex manifolds called the Chern-Ricci flow.The Kähler-Ricci flow started as a small branch of the study of Hamilton's Ricci flow, but by now is itself a vast area of research. As a consequence, we have had to omit many topics. For the interested reader seeking more complete expository sources: the chapter [66] by Jian Song and the author contains many of the results of these notes and much more; the works [4,8,31] (in the same volume as [66]) and the more general survey [53] are excellent sources of information.The author thanks Matt Gill who was the teaching assistant for this course, for his help in writing the exercises. In addition, thanks go to the organizers, Hubert Bray, Greg Galloway, Rafe Mazzeo and Natasa Sesum, of the research program of the 2013 PCMI Summer Session for giving the author the opportunity to participate in this exciting event. Discussions with researchers and graduate students at the Park City Math Institute were invaluable in shaping the form of these notes. The author also thanks Valentino Tosatti for some helpful comments on a previous version of these notes.

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“…Higher dimensional metric surgeries via the Kähler-Ricci flow are constructed for certain families of projective manifolds in [33,24]. For more details of the behavior of the Kähler-Ricci flow with finite time singularity, see [41,26,8,25,42,9,10,39].…”

Section: Introductionmentioning

confidence: 99%

Scalar V-soliton equation and Kähler-Ricci flow on symplectic quotients

Li¹

2020

Preprint

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In this paper, we consider the V -soliton equation which is a degenerate fully nonlinear equation introduced by La Nave and Tian in their work on Kähler-Ricci flow on symplectic quotients. One can apply the interpretation to study finite time singularities of the Kähler-Ricci flow. As in the case of Kähler-Einstein metrics, we can also reduce the V -soliton equation to a scalar equation on Kähler potentials, which is of Monge-Ampère type. We formulate some preliminary estimates for such a scalar equation on a compact Kähler manifold M .

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Kähler-Ricci flow on projective bundles over Kähler-Einstein manifolds

Cited by 11 publications

References 19 publications

On a twisted conical Kähler–Ricci flow

On a twisted conical Kähler–Ricci flow

The Kähler-Ricci flow on compact Kähler manifolds

Scalar V-soliton equation and Kähler-Ricci flow on symplectic quotients

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