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

TL;DR

This paper proves that the maximum Fano index for canonical weak Fano threefolds is 66, confirming a conjecture and introducing new formulas and methods for analyzing singularities and geometric properties of these varieties.

Contribution

It establishes the optimal upper bound of 66 for the Fano index of canonical weak Fano 3-folds, solving a conjecture and developing new tools for their study.

Findings

Fano index of canonical weak Fano 3-folds is at most 66

New Riemann–Roch formula for canonical 3-folds

Detailed analysis of non-isolated singularities

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.