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Holomorphic fibrations on Calabi-Yau manifolds and collapsing

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

March 21, 2016 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Valentino Tosatti (New York University, Courant Institute)
Location: MSRI: Simons Auditorium
  • mathematical physics

  • complex differential geometry

  • Kahler metric

  • mirror symmetry

  • Calabi-Yau manifold

  • Ricci curvature

  • Ricci flatness

  • algebraic geometry and GAGA

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



Consider a compact Calabi-Yau manifolds with a holomorphic fibration onto a lower-dimensional base. Pulling back a Kahler class from the base, we obtain a class on the boundary of the Kahler cone, which is a limit of Kahler classes. These classes contain Ricci-flat metrics, which in the limit collapse to a twisted Kahler-Einstein metric on the base (away from the singular fibers). Furthermore if we rescale so that the fibers have fixed size, then away from the singular fibers the limit is a cylinder over a Ricci-flat fiber. This is based on joint work with Weinkove and Yang, with Hein and with Zhang, and is directly related to the topic of the talk by Mark Gross

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