Optimal upper bound for degrees of canonical Fano threefolds of Picard number one

TL;DR

This work determines the sharp upper bound for the anti-canonical degree of $\mathbb{Q}$-factorial canonical Fano $3$-folds with Picard number one, proving $(-K_X)^3 \le 72$ with equality only for $X \cong \mathbb{P}(1,1,1,3)$ or $\mathbb{P}(1,1,4,6)$. The authors build a Kawamata–Miyaoka type inequality relating $(-K_X)^3$ to the generalized second Chern class $\hat{c}_2(X)$, and extend the framework to $\epsilon$-lc Fano varieties using a $\,\mathbb{Q}$-variant of Langer’s inequality and foliation techniques to control stability. They derive explicit bounds depending on the $\mathbb{Q}$-Fano index and distinguish Gorenstein from non-Gorenstein cases, ultimately completing the degree classification for this class of Fano $3$-folds. The results sharpen prior bounds and illuminate how orbifold Chern data constrain the geometry of singular Fano varieties, contributing to the broader program of classifying Fano $3$-folds with mild singularities.

Abstract

We show that for a $\mathbb Q$-factorial canonical Fano $3$-fold $X$ of Picard number $1$, $(-K_X)^3\leq 72$. The main tool is a Kawamata--Miyaoka type inequality which relates $(-K_X)^3$ with $\hat{c}_2(X)\cdot c_1(X)$, where $\hat{c}_2(X)$ is the generalized second Chern class.

Theorems & Definitions (49)

• Theorem 1.1: =Theorem \ref{['thm.sharpboundofdegree']}
• Theorem 1.2
• Lemma 2.1
• proof
• Theorem 2.2: Reid1987
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof