Slopes of smooth curves on Fano manifolds

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 Kahler metric. This paper presents a study of slope stability of Fano manifolds of dimension n >= 3 with respect to smooth curves. The question turns out to be easy for curves of genus at least 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 three-fold X with Picard number 1 is slope stable with respect to every smooth curve unless X is the projective space.