Global Ricci Curvature Behaviour for the K\"ahler-Ricci Flow with Finite Time Singularities

TL;DR

This paper studies the behavior of Ricci curvature under the K"ahler-Ricci flow on compact manifolds with finite-time singularities, providing new curvature estimates under specific geometric conditions.

Contribution

It establishes $L^4$-like estimates for Ricci curvature and classifies the Riemannian curvature as Type I in the $L^2$-sense for the flow with finite-time singularities.

Findings

Proves an $L^4$-like estimate on Ricci curvature behavior.

Shows the Riemannian curvature is Type I in the $L^2$-sense.

Applies to cases where the initial cohomology class is rational.

Abstract

We consider the K\"ahler-Ricci flow $(X, \omega(t))_{t \in [0,T)}$ on a compact manifold where the time of singularity, $T$, is finite. We assume the existence of a holomorphic map from the K\"ahler manifold $X$ to some analytic variety $Y$ which admits a K\"ahler metric on a neighbourhood of the image of $X$ and that the pullback of this metric yields the limiting cohomology class along the flow. This is satisfied, for instance, by the assumption that the initial cohomology class is rational, i.e., $[\omega_0] \in H^{1,1}(X,\mathbb{Q})$. Under these assumptions we prove an $L^4$-like estimate on the behaviour of the Ricci curvature and that the Riemannian curvature is Type $I$ in the $L^2$-sense.