On fibration stability after Dervan-Sektnan and singularities

TL;DR

This paper introduces a new stability concept called $$-stability for fibrations, demonstrating that $$-semistability ensures semi log canonical singularities and constrains their centers on Fano fibrations.

Contribution

It proposes $$-stability as a modification of existing fibration stability and establishes its implications for singularities and their centers in Fano fibrations.

Findings

$$-semistable fibrations have semi log canonical singularities.

$$-stability restricts semi log canonical centers on Fano fibrations.

The new stability concept refines understanding of singularities in fibrations.

Abstract

We introduce $\mathfrak{f}$-stability, a modification of fibration stability of Dervan-Sektnan [12], and show that $\mathfrak{f}$-semistable fibrations have only semi log canonical singularities. Moreover, $\mathfrak{f}$-stability puts restrictions on semi log canonical centers on Fano fibrations.