A canonical Fano threefold has Fano index $\leq 66$

TL;DR

The paper introduces a sharp upper bound $q_{ ext{Q}}(X)\le 66$ for the $ ext{Q}$-Fano index of canonical weak Fano $3$-folds, resolving a conjecture of Wang in dimension three. To achieve this, it develops a novel RR framework for canonical $3$-folds via sequential terminalizations and Weil pullbacks, coupled with a Kawamata–Miyaoka type inequality and a detailed study of crepant curves and their intersections. It leverages integrality constraints on RR expressions to enumerate 36 potential numerical types and then systematically rules them out, employing geometry of foliations of rank two to handle stubborn cases. The results provide a definitive, optimal bound and introduce robust techniques that may aid future classifications of canonical Fano varieties and related invariants.

Abstract

We show that the $\mathbb{Q}$-Fano index of a canonical weak Fano $3$-fold is at most $66$. This upper bound is optimal and gives an affirmative answer to a conjecture of Chengxi Wang in dimension $3$. During the proof, we establish a new Riemmann--Roch formula for canonical $3$-folds and provide a detailed study of non-isolated singularities on canonical Fano $3$-folds, concerning both their local and global properties. Our proof also involves a Kawamata--Miyaoka type inequality and geometry of foliations of rank $2$ on canonical Fano $3$-folds.

Theorems & Definitions (120)

• Theorem 1.1
• Example 1.2: kasprzyk2*Table 3
• Theorem 2.1: jiang-liu-liu*Theorem 4.2
• Definition 2.2
• Definition 2.3: Non-split crepant divisor
• Lemma 2.4
• proof
• Remark 2.5
• Theorem 2.6: jiang-liu-liu*Theorem 3.8
• proof