Bergman iteration and C^(∞)-convergence towards Kähler-Ricci flow

Ryosuke Takahashi

Paper · Jun 9, 2016

Bergman iteration and $C^{\infty}$ -convergence towards Kähler-Ricci flow

Ryosuke Takahashi

math.DGarXiv:1606.03019v5

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Abstract

On a polarized manifold $(X, L)$ , the Bergman iteration $φ_{k}^{(m)}$ is defined as a sequence of Bergman metrics on $L$ with two integer parameters $k, m$ . We study the relation between the Kähler-Ricci flow $φ_{t}$ at any time $t \geq 0$ and the limiting behavior of metrics $φ_{k}^{(m)}$ when $m = m (k)$ and the ratio $m / k$ approaches to $t$ as $k \to \infty$ . Mainly, three settings are investigated: the case when $L$ is a general polarization on a Calabi-Yau manifold $X$ and the case when $L = \pm K_{X}$ is the (anti-) canonical bundle. Recently, Berman showed that the convergence $φ_{k}^{(m)} \to φ_{t}$ holds in the $C^{0}$ -topology, in particular, the convergence of curvatures holds in terms of currents. In this paper, we extend Berman's result and show that this convergence actually holds in the smooth topology.

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Submitted: Jun 9, 2016
Last revised: Mar 13, 2019 (v5)
Subjects: math.DG
Journal: Osaka J. Math. 4 (2018), Volume 55, 713-729

Comments: 17 pages, final version

Ricci flow Kähler–Ricci flow Ricci-flat & Calabi–Yau Kähler geometry

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.
5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

[1] 5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.

[2] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[3] 10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

Bergman iteration and $C^{\infty}$ -convergence towards Kähler-Ricci flow

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Bergman iteration and C∞-convergence towards Kähler-Ricci flow

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Bergman iteration and $C^{\infty}$ -convergence towards Kähler-Ricci flow