Collapsing of abelian fibered calabi-yau manifolds

Mark Gross*, Valentino Tosatti, Yuguang Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Scopus citations

Abstract

We study the collapsing behavior of Ricci-flat Kähler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkähler manifolds in the large complex structure limit. In this setting, we prove an analogue of a result of Gross and Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.

Original languageEnglish (US)
Pages (from-to)517-551
Number of pages35
JournalDuke Mathematical Journal
Volume162
Issue number3
DOIs
StatePublished - Feb 2013

ASJC Scopus subject areas

  • Mathematics(all)

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