Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On slope unstable Fano varieties

Yen-An Chen, Ching-Jui Lai

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-Wiśniewski 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 Kähler-Einstein metric, yet whose tangent sheaf is slope unstable.

On slope unstable Fano varieties

Abstract

For Fano varieties, significant progress has been made recently in the study of -stability, while the understanding of the weaker but more algebraic concept of -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-Wiśniewski 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 -slope unstable weak del Pezzo surfaces with canonical singularities. As by-products, we provide the first conceptual proof that and are the only -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 Kähler-Einstein metric, yet whose tangent sheaf is slope unstable.
Paper Structure (13 sections, 22 theorems, 17 equations, 4 figures)

This paper contains 13 sections, 22 theorems, 17 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Sketch of the proof of Theorem \ref{['thm:nonsingular-wdP']}
  3. Further discussion
  4. Preliminaries
  5. Foliations
  6. Log canonical foliated surface singularities
  7. Toric varieties and toric foliations
  8. A foliated MMP on an unstable weak Fano variety
  9. Maximal destabilizing sheaf
  10. A foliated MMP
  11. Weak del Pezzo surfaces
  12. dim Z+=1: The classification
  13. dim Z+=0: Non-existence

Key Result

Theorem 1.7

Let $X$ be a nonsingular weak del Pezzo surface. If the tangent bundle $T_X$ is $(-K_X)$-slope unstable, then $X$ is either the Hirzebruch surface $\mathbb{F}_n$ with $n\in\{0,1,2\}$, or $X\cong X_{n,1}, X_{n,2},$ or $X_{n,3}$ as described in Examples eg:one_2blowup, eg:two_2blowup, and eg:three_2bl

Figures (4)

  • Figure 1: A curve configuration of Example \ref{['eg:three_2blowup']}. The horizontal line indicates the proper transform of the negative section on $\mathbb{F}_n$, the top six lines indicate the exceptional curves, and the bottom three skew lines indicate the strict transform of the fiber of the canonical fibration on $\mathbb{F}_n$.
  • Figure 2: A visualization of a singular del Pezzo surface $X$ with $\operatorname{Sing}(X)=6A_1$ admitting a weak KE metric but whose tangent sheaf is slope unstable. This is obtained by contracting all $(-2)$-curves of Example \ref{['eg:two_2blowup']} when $n=1$.
  • Figure 3: $\Sigma_0(1)=\{\rho_1,\rho_2,\rho_3,\rho_4\}$; $\Sigma_2(1)=\{\rho_i:1\leq i\leq 6\}$.
  • Figure 4: The visualization of Case (no. 8c) when $L_0^2=2$.

Theorems & Definitions (50)

  • Conjecture 1.1
  • Definition 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • Corollary 1.11
  • ...and 40 more