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arXiv · 2508.16364

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

Chen Jiang·Haidong Liu
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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.

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Chen Jiang, Haidong Liu. 2025-08-22. A canonical Fano threefold has Fano index $\leq 66$. https://doi.org/10.1112/plms.70195

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