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

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.

50 pages, 5 tables. Comments are welcome! This preprint supersedes our old preprint arXiv:2505.19541. Ver2: We clarify an inaccurate statement on crepant divisors in Ver1 pointed out by Professor Yuri Prokhorov and introduce the concept of non-split crepant divisors (Definition 2.3), main results are not affected. Ver3: The published version