KAWA lecture notes on the Kähler–Ricci flow

Tosatti, Valentino

doi:10.5802/afst.1571

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Introduction2

A Flow and Nonlinear Semistable1

Setup and Notation1

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2019

2024

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Cited by 43 publications

(34 citation statements)

References 84 publications

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“…[18, Conjecture 6.7] For the Kähler-Ricci flow on any compact Kähler manifold with nef canonical line bundle, the singularity type at infinity does not depend on the choice of the initial metric.…”

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confidence: 99%

Infinite-Time Singularity Type of the Kähler–Ricci Flow

Zhang

2017

J Geom Anal

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mentioning

confidence: 99%

Infinite-Time Singularity Type of the Kähler–Ricci Flow

Zhang

2017

J Geom Anal

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“…In this subsection, we will study some properties of the flow (2.5) by using the method of studying Kähler-Ricci flow (see e.g. [10,34]).…”

Section: A Flow and Nonlinear Semistablementioning

confidence: 99%

Remarks on the geodesic-Einstein metrics of a relative ample line bundle

Wan¹,

Wang

2020

Math. Z.

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Let π : X → M be a holomorphic fibration over a compact complex manifold M with compact fibers, L be a relative ample line bundle over X . In this paper, we prove that the pair (X , L) is nonlinear semistable if the Donaldson type functional L(·, ψ) is bounded from below and the long time existence of the flow (0.1). For the case of trωc(φ) ≥ 0, we give a sufficient condition deg ω π * (L + K X /M ) = 0 for the existence of geodesic-Einstein metric. We also introduce the definitions of S-class and C-class inspired by the Segre class and Chern class, and find two inequalities in terms of S-forms and C-forms if there exists a geodesic-Einstein metric. If L is ample, the positivity of S-forms is proved. If moreover, the line bundle L admits a geodesic-Einstein metric and dim M = 2, the second C-class C2(L) is proved to be numerical positive. We also prove the equivalence between the geodesic-Einstein metrics on a relative ample line bundle and the Hermitian-Einstein metrics on a quasi-vector bundle A 0,0 . Lastly, we discuss some examples on geodesic-Einstein metrics.

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“…This definition gives a well-defined metric (see e.g. [52]). In our case, we may take Ψy to be Ω(∂y, ·, ·) for a local coordinate y on Σ such that ϕ * ∂ ζ = dy.…”

Section: Setup and Notationmentioning

confidence: 99%

The Anomaly Flow over Riemann Surfaces

Teng¹,

Huang

Picard

2019

International Mathematics Research Notices

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We initiate the study of a new nonlinear parabolic equation on a Riemann surface. The evolution equation arises as a reduction of the Anomaly flow on a fibration. We obtain a criterion for long-time existence for this flow, and give a range of initial data where a singularity forms in finite time, as well as a range of initial data where the solution exists for all time. A geometric interpretation of these results is given in terms of the Anomaly flow on a Calabi-Yau threefold.

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KAWA lecture notes on the Kähler–Ricci flow

Abstract: These lecture notes provide an introduction to the study of the Kähler-Ricci flow on compact Kähler manifolds, and a detailed exposition of some recent developments.

Cited by 43 publications

References 84 publications

Infinite-Time Singularity Type of the Kähler–Ricci Flow

Infinite-Time Singularity Type of the Kähler–Ricci Flow

Remarks on the geodesic-Einstein metrics of a relative ample line bundle

The Anomaly Flow over Riemann Surfaces

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