A canonical Fano threefold has Fano index $\leq 66$
Chen Jiang, Haidong Liu

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 -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.
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