On slope unstable Fano varieties

TL;DR

This paper investigates the algebraic stability of Fano varieties, especially tangent bundle slope stability, using modern minimal model program techniques, leading to classifications and new phenomena in low dimensions.

Contribution

It introduces a method to analyze the geometry of destabilizing sheaves of Fano varieties, providing classifications for weak del Pezzo surfaces and revealing novel stability phenomena.

Findings

Complete classification of $(-K)$-slope unstable weak del Pezzo surfaces with canonical singularities.

Proof that $P^1 imes P^1$ and $F_1$ are the only $(-K)$-slope unstable nonsingular del Pezzo surfaces.

Existence of a del Pezzo surface with A-type singularities that admits a weak Kähler-Einstein metric but has slope unstable tangent sheaf.

Abstract

For Fano varieties, significant progress has been made recently in the study of $K$-stability, while the understanding of the weaker but more algebraic concept of $(-K)$-slope stability remains intricate. For instance, a conjecture attributed to Iskovskikh states that the tangent bundle of a Picard rank one Fano manifold is slope stable. Peternell-Wi\'sniewski and Hwang proved this conjecture up to dimension five in 1998, but Kanemitsu later disproved it in 2021. To address this gap in understanding, we present a method that aims to characterize the geometry associated with the maximal destabilizing sheaf of the tangent sheaf of a Fano variety. This approach utilizes modern advancements in the foliated minimal model program. In dimension two, our approach leads to a complete classification of $(-K)$-slope unstable weak del Pezzo surfaces with canonical singularities. As by-products, we provide the first conceptual proof that $\mathbb{P}^1 \times \mathbb{P}^1$ and $\mathbb{F}_1$ are the only $(-K)$-slope unstable nonsingular del Pezzo surfaces, recovering a classical result of Fahlaoui in 1989. We also uncover a phenomenon that does not occur for Fano manifolds: there exists a del Pezzo surface with type A singularities admitting a weak K\"ahler-Einstein metric, yet whose tangent sheaf is slope unstable.