The K"ahler-Ricci flow with Log Canonical Singularities

TL;DR

This paper proves the existence and convergence of the Kähler-Ricci flow on projective varieties with log canonical singularities, extending previous results to more general singularities and including the construction of flow solutions involving divisorial contractions and flips.

Contribution

It generalizes the existence of the Kähler-Ricci flow to varieties with log canonical singularities and demonstrates convergence to Kähler-Einstein metrics, also constructing flow solutions with complex birational transformations.

Findings

Flow exists on varieties with log canonical singularities.

Normalized flow converges to Kähler-Einstein metrics with negative Ricci curvature.

Constructs solutions performing divisorial contractions and flips.

Abstract

We establish the existence of the K"ahler-Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song-Tian \cite{ST3} in case of projective varieties with klt singularities. We also prove that the normalized K"ahler-Ricci flow will converge to the \ka-Einstein metric with negative Ricci curvature on semi-log canonical models in the sense of currents. Finally we also construct K"ahler-Ricci flow solutions performing divisorial contractions and flips with log canonical singularities.