A canonical Fano threefold has Fano index $\leq 66$
Abstract
We show that the -Fano index of a canonical weak Fano -fold is at most . This upper bound is optimal and gives an affirmative answer to a conjecture of Chengxi Wang in dimension . During the proof, we establish a new Riemmann--Roch formula for canonical -folds and provide a detailed study of non-isolated singularities on canonical Fano -folds, concerning both their local and global properties. Our proof also involves a Kawamata--Miyaoka type inequality and geometry of foliations of rank on canonical Fano -folds.
Cite
@article{arxiv.2508.16364,
title = {A canonical Fano threefold has Fano index $\leq 66$},
author = {Chen Jiang and Haidong Liu},
journal= {arXiv preprint arXiv:2508.16364},
year = {2025}
}
Comments
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