A canonical Fano threefold has Fano index ⩽66

Abstract

We show that the ‐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 Riemann–Roch formula for canonical 3‐folds and provide a detailed study of nonisolated 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.