Slopes of smooth curves on Fano manifolds

TL;DR

This paper classifies when Fano manifolds are slope stable with respect to smooth curves, especially focusing on rational curves, and finds that most are stable except for projective space.

Contribution

It provides a complete classification of slope stability for Fano manifolds concerning smooth curves, highlighting the special case of projective space.

Findings

Fano manifolds of dimension ≥3 are mostly slope stable with respect to smooth curves.

Fano threefolds with Picard number 1 are slope stable unless they are projective space.

The case of rational curves is particularly significant in the stability analysis.

Abstract

Ross and Thomas introduced the concept of slope stability to study K-stability, which has conjectural relation with the existence of constant scalar curvature K\"ahler metric. This paper presents a study of slope stability of Fano manifolds of dimension $n\geq 3$ with respect to smooth curves. The question turns out to be easy for curves of genus $\geq 1$ and the interest lies in the case of smooth rational curves. Our main result classifies completely the cases when a polarized Fano manifold $(X, -K_X)$ is not slope stable with respect to a smooth curve. Our result also states that a Fano threefold $X$ with Picard number 1 is slope stable with respect to every smooth curve unless $X$ is the projective space.