The kähler-ricci flow, ricci-flat metrics and collapsing limits

Valentino Tosatti; Ben Weinkove; Xiaokui Yang

doi:10.1353/ajm.2018.0016

The kähler-ricci flow, ricci-flat metrics and collapsing limits

Valentino Tosatti, Ben Weinkove, Xiaokui Yang

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate the Kähler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kähler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kähler metrics.

Original language	English (US)
Pages (from-to)	653-698
Number of pages	46
Journal	American Journal of Mathematics
Volume	140
Issue number	3
DOIs	https://doi.org/10.1353/ajm.2018.0016
State	Published - Jun 2018

ASJC Scopus subject areas

General Mathematics

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10.1353/ajm.2018.0016

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title = "The k{\"a}hler-ricci flow, ricci-flat metrics and collapsing limits",

abstract = "We investigate the K{\"a}hler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat K{\"a}hler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat K{\"a}hler metrics.",

author = "Valentino Tosatti and Ben Weinkove and Xiaokui Yang",

note = "Funding Information: Research supported in part byNSF grants DMS-1308988, DMS-1332196, and DMS-1406164; research of the first author supported in part by a Sloan Research Fellowship; research of the third author supported by China{\textquoteright}s Recruitment Program and NSFC 11688101. Publisher Copyright: {\textcopyright} 2018 by Johns Hopkins University Press.",

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N2 - We investigate the Kähler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kähler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kähler metrics.

AB - We investigate the Kähler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kähler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kähler metrics.

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The kähler-ricci flow, ricci-flat metrics and collapsing limits

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