Geometry & Topology

Infinite-time singularities of the Kähler–Ricci flow

Valentino Tosatti and Yuguang Zhang

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Abstract

We study the long-time behavior of the Kähler–Ricci flow on compact Kähler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so it is fibered by Calabi–Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and a flat metric on Euclidean space. An analogous result holds for collapsing limits of Ricci-flat Kähler metrics.

Article information

Source
Geom. Topol., Volume 19, Number 5 (2015), 2925-2948.

Dates
Received: 31 August 2014
Revised: 16 November 2014
Accepted: 15 December 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858852

Digital Object Identifier
doi:10.2140/gt.2015.19.2925

Mathematical Reviews number (MathSciNet)
MR3416117

Zentralblatt MATH identifier
1328.53089

Subjects
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx] 58J35: Heat and other parabolic equation methods

Keywords
Kähler–Ricci flow infinite-time singularity Calabi–Yau manifold collapsing

Citation

Tosatti, Valentino; Zhang, Yuguang. Infinite-time singularities of the Kähler–Ricci flow. Geom. Topol. 19 (2015), no. 5, 2925--2948. doi:10.2140/gt.2015.19.2925. https://projecteuclid.org/euclid.gt/1510858852


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