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Gromov-Hausdorff collapse of Calabi-Yau manifolds

Kähler Geometry, Einstein Metrics, and Generalizations March 21, 2016 - March 25, 2016

March 24, 2016 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): Mark Gross (University of Cambridge)
Location: MSRI: Simons Auditorium
  • algebraic geometry and GAGA

  • complex differential geometry

  • mathematical physics

  • Kahler metric

  • mirror symmetry

  • Gromov-Hausdorff metric

  • abelian varieties

  • Ricci flatness

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC



I will talk about joint work with Tosatti and Zhang on Gromov-Hausdorff collapse of abelian fibred Calabi-yau manifolds. One considers a Calabi-Yau manifold M with a holomorphic map M->N with general fibres being abelian varieties. Given a suitably chosen sequence of Ricci-flat metrics so that the volume of the fibres goes to zero, we show that the Kahler form on M converges to the pull-back of a form on N. Stronger results if the dimension of N is 1 imply that in fact M converges in the Gromov-Hausdorff sense to N. This leads to an extension of an old collapsing result of myself and Wilson in the K3 case with no assumptions on the types of singular fibres

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