On the Ricci curvature of Kahler-Ricci Flow

TL;DR

This paper studies the behavior of Ricci curvature under the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical bundles, showing convergence to a generalized Kähler-Einstein metric in certain cases.

Contribution

It demonstrates the convergence of Ricci curvature to a generalized Kähler-Einstein metric for manifolds with Kodaira dimension one under the Kähler-Ricci flow.

Findings

Ricci curvature converges to negative of generalized Kähler-Einstein metric

Convergence occurs locally away from singular sets

Results apply to manifolds with semi-ample canonical bundle

Abstract

In this paper, we consider $n$-dimensional compact K$\ddot{a}$hler manifold with semi-ample canonical line bundle under the long time solution of K$\ddot{a}$hler Ricci Flow. In particular, if the Kodaira dimension is one, Ricci curvature converge to negative of generalized K$\ddot{a}$hler Einstein metric $\omega_B$ locally away from singular set in $C^0_{loc}(\omega(t))$ topology.