Towards a criterion for slope stability of Fano manifolds along divisors

TL;DR

This paper introduces a simple criterion for assessing slope stability of Fano manifolds along divisors, with applications to classifying stability and counterexamples to Aubin's conjecture, especially focusing on Fano threefolds.

Contribution

It provides a new, straightforward criterion for slope stability of Fano manifolds along divisors and offers a complete classification for Fano threefolds.

Findings

Fano manifolds are slope stable along an ample divisor unless they are projective spaces.

Counterexamples to Aubin's conjecture are constructed.

Complete classification of slope stability for Fano threefolds.

Abstract

We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a projective space and $D$ is a hyperplane section. We also give counterexamples to Aubin's conjecture on the relation between the anticanonical volume and the existence of a K\"ahler-Einstein metric. Finally, we consider the case that $\dim X=3$; we give a complete answer for slope (semi)stability along divisors of Fano threefolds.