Singular cscK metrics on smoothable varieties

TL;DR

This paper proves the semi-continuity of the coercivity threshold of the Mabuchi functional and establishes the existence of singular cscK metrics on smoothable klt varieties as limits of metrics on nearby fibers.

Contribution

It introduces a new strong topology in pluripotential theory for families and demonstrates the existence of singular cscK metrics on smoothable klt varieties.

Findings

Lower semi-continuity of the coercivity threshold established.

Existence of singular cscK metrics on smoothable klt varieties proven.

Uniform estimates for cscK metrics developed.

Abstract

We prove the lower semi-continuity of the coercivity threshold of Mabuchi functional along a degenerate family of normal compact K\"ahler varieties with klt singularities. Moreover, we establish the existence of singular cscK metrics on $\mathbb{Q}$-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive, these arise as a limit of cscK metrics on close-by fibres. The proof relies on developing a novel strong topology of pluripotential theory in families and establishing uniform estimates for cscK metrics.