Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow

Abstract

We prove a uniform diameter bound for long time solutions of the normalized Kahler-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 $X_{can}$. 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. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to its canonical model when $X$ has Kodaira dimension $1$ with $K_X$ being semi-ample and the general fibre of $X$ over its canonical model being a complex torus. We also prove the Gromov-Hausdorff limit of collapsing Ricci-flat Kahler 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.