Higher-order estimates of long-time solutions to the Kähler-Ricci flow

In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. By developing sharp new parabolic Schauder estimates on cylinders, we proved that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges in C_loc^∞-topology away from singular fibers to a negative Kähler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fibers are biholomorphic to each other.