Limits of Calabi-Yau metrics when the Kähler class degenerates

Valentino Tosatti

doi:10.4171/JEMS/165

Limits of Calabi-Yau metrics when the Kähler class degenerates

Valentino Tosatti

Mathematics

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

Abstract

We study the behavior of families of Ricci-flat Kähler metrics on a projective Calabi-Yau manifold when the Kähler classes degenerate to the boundary of the ample cone.We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry.

Original language	English (US)
Pages (from-to)	755-776
Number of pages	22
Journal	Journal of the European Mathematical Society
Volume	11
Issue number	4
DOIs	https://doi.org/10.4171/JEMS/165
State	Published - 2009

Keywords

Calabi-Yau manifolds
Degenerate complex Monge-Ampère equations
Ricci-flat metrics

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.4171/JEMS/165

Cite this

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abstract = "We study the behavior of families of Ricci-flat K{\"a}hler metrics on a projective Calabi-Yau manifold when the K{\"a}hler classes degenerate to the boundary of the ample cone.We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry.",

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AB - We study the behavior of families of Ricci-flat Kähler metrics on a projective Calabi-Yau manifold when the Kähler classes degenerate to the boundary of the ample cone.We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry.

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KW - Degenerate complex Monge-Ampère equations

KW - Ricci-flat metrics

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Limits of Calabi-Yau metrics when the Kähler class degenerates

Abstract

Keywords

ASJC Scopus subject areas

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