## Differential Geometry Seminar: Collapsing behavior of Ricci-flat Kähler metrics and long time solutions of the Kähler-Ricci flow

Seminar | February 10 | 3:10-4 p.m. | 939 Evans Hall

Jian Song, Rutgers

Department of Mathematics

We prove a uniform diameter bound for long time solutions of the normalized Kähler-Ricci flow on an n-dimensional projective manifold X with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of X over its canonical model. This assumption on the Ricci curvature always holds when the Kodaira dimension of X is n, n-1 or when the general fibre of X over its canonical model is a complex torus. We also prove that the Gromov-Hausdorff limit of collapsing Ricci-flat Kähler metrics on a holomorphically fibred Calabi-Yau manifold is unique and is homeomorphic to the metric completion of the corresponding twisted Kahler-Einstein metric on the regular part of its base.

lott@math.berkeley.edu