Collapsing immortal Kähler-Ricci flows
Hans-Joachim Hein, Man-Chun Lee, Valentino Tosatti
TL;DR
The paper analyzes the normalized Kähler-Ricci flow on compact Kähler manifolds with nef, semiample canonical bundle and intermediate Kodaira dimension, showing collapse to a canonical base metric on the Iitaka fibration. It develops a parabolic shrinking Hölder framework with a fiberwise connection to control the degenerate ellipticity, and proves a Selection Theorem to construct a finite obstruction hierarchy; through an intricate blow-up/contradiction analysis across three regimes, it establishes a precise parabolic asymptotic expansion for the evolving metrics. The main outcome is a full higher-order convergence result away from singular fibers and a uniform Ricci curvature description Ric$( ext{ω}^ullet(t))=- ext{ω}_{ m can}+ ext{Err}$ with $ ext{Err} o 0$, thereby resolving Song–Tian conjectures in this intermediate Kodaira dimension setting. The methods extend collapsing Calabi–Yau metric techniques to the parabolic setting and yield sharp regularity and curvature information, with potential implications for the structure of minimal models and the broader Kähler–Ricci flow program.
Abstract
We consider the Kähler-Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian.