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arXiv · 1005.4310

Slopes of smooth curves on Fano manifolds

Jun-Muk Hwang·Hosung Kim·Yongnam Lee·Jihun Park
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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 Kähler 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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Jun-Muk Hwang, Hosung Kim, Yongnam Lee, Jihun Park. 2011-03-10. Slopes of smooth curves on Fano manifolds. https://doi.org/10.1112/blms%2Fbdr020

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