Slopes of smooth curves on Fano manifolds

Details Slopes of smooth curves on Fano manifolds Slopes of smooth curves on Fano manifolds

2010-05-24

arxiv.org/abs/1005.4310v3

Mathematics

Algebraic Geometry

13 pages, Theorems in the original version were modified. This paper will be published in the Bulletin of the London Mathemati

Scientific paper

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.

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