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Infinite-time singularities of the Kähler–Ricci flow

Valentino Tosatti and Yuguang Zhang

Geometry & Topology 19 (2015) 2925–2948

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.

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Kähler–Ricci flow, infinite-time singularity, Calabi–Yau manifold, collapsing
Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 53C55, 58J35
Received: 31 August 2014
Revised: 16 November 2014
Accepted: 15 December 2014
Published: 20 October 2015
Proposed: John Lott
Seconded: Tobias H Colding, Gang Tian
Valentino Tosatti
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208
Yuguang Zhang
Yau Mathematical Sciences Center
Tsinghua University
Beijing 100084