A Kawamata--Miyaoka type inequality for Fano varieties of arbitrary Picard number

TL;DR

The paper proves a Kawamata--Miyaoka type inequality for $\mathbb Q$-factorial canonical weak Fano varieties with $q_{\mathbb Q}(X)\ge 3$, showing $c_1(X)^n \le 4\,\hat{c}_2(X)\cdot c_1(X)^{n-2}$ and that $4\,\hat{c}_2(X)-c_1(X)^2$ is pseudoeffective; this is established by analyzing the foliation induced by the maximal destabilizing subsheaf of $\mathcal T_X$ and by applying a $\mathbb{Q}$-Bogomolov--Gieseker type inequality. The result extends to varieties with arbitrary Picard number and various singularity types, yielding a broad toolkit for bounding numerical invariants via foliation methods. As an application, the authors bound the $\mathbb Q$-Fano index of Gorenstein canonical Fano $3$-folds, showing $q_{\mathbb Q}(X)\in\{m\le 22\}\cup\{24,30,42\}$, and relate this to explicit realizations by weighted projective spaces. These findings contribute to the classification program for Fano varieties with mild singularities by linking stability, Chern class inequalities, and index bounds.

Abstract

Let $X$ be a $\mathbb Q$-factorial canonical weak Fano variety of dimension $n\geq 2$. We show that if the $\mathbb Q$-Fano index $q_{\mathbb Q}(X)\geq 3$, then $X$ satisfies a Kawamata--Miyaoka type inequality: \[c_1(X)^n\leq 4\,\hat c_2(X)\cdot c_1(X)^{n-2}.\] As an application, we show that the $\mathbb Q$-Fano index of a Gorenstein canonical Fano $3$-fold lies in the set $\{m\in\mathbb Z_{>0}\mid m\leq 22\} \cup\{24,30,42\}$.

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3: Corollary \ref{['cor.index']}
• Proposition 2.1
• proof
• Theorem 2.2: jiang-liu-liu*Theorem 4.2
• Theorem 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 3.1