Infinite-time singularities of the Kähler–Ricci flow

Tosatti, Valentino¹ ; Zhang, Yuguang²

¹ Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA
² Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

Geometry & topology, Tome 19 (2015) no. 5, pp. 2925-2948.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.2925

Classification : 53C44, 53C55, 58J35
Keywords: Kähler–Ricci flow, infinite-time singularity, Calabi–Yau manifold, collapsing

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