Convergence of cscK metrics on smooth minimal models of general type

TL;DR

This paper proves that sequences of constant scalar curvature Kähler metrics on smooth minimal models of general type converge smoothly away from a subvariety to the canonical Kähler-Einstein metric, confirming a conjecture about their convergence behavior.

Contribution

It establishes the partial convergence of cscK metrics to the canonical Kähler-Einstein metric on minimal models of general type, advancing understanding of their geometric limits.

Findings

Sequences of cscK metrics converge smoothly away from a subvariety.

The convergence is towards the singular Kähler-Einstein metric.

The result confirms a conjecture by Jian-Shi-Song.

Abstract

We consider constant scalar curvature K\"{a}hler metrics on a smooth minimal model of general type in a neighborhood of the canonical class, which is the perturbation of the canonical class by a fixed K\"{a}hler metric. We show that sequences of such metrics converge smoothly on compact subsets away from a subvariety to the singular K\"{a}hler Einstein metric in the canonical class. This confirms partially a conjecture of Jian-Shi-Song about the convergence behavior of such sequences.