Adiabatic limits of ricci-flat kähler metrics

Valentino Tosatti*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

43 Scopus citations

Abstract

We study adiabatic limits of Ricci-flat Kähler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Amp`ere equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metric has Ricci curvature equal to a Weil- Petersson metric that measures the variation of complex structure of the Calabi-Yau fibers. This generalizes results of Gross-Wilson for K3 surfaces to higher dimensions.

Original languageEnglish (US)
Pages (from-to)427-453
Number of pages27
JournalJournal of Differential Geometry
Volume84
Issue number2
DOIs
StatePublished - 2010

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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