Optimal upper bound for degrees of canonical Fano threefolds of Picard number one

Abstract

We show that for a $\mathbb{Q}$-factorial canonical Fano $3$-fold $X$ of Picard number $1$, $(-K_X{)}^3\le 72$. The main tool is a Kawamata--Miyaoka type inequality which relates $(-K_X{)}^3$ with ${\hat{c}}_2(X)\cdot c_1(X)$, where ${\hat{c}}_2(X)$ is the generalized second Chern class.

16 pages, 4 tables. Comments are welcome! Ver2: 17 pages, 5 tables. Ver3: 19 pages, minor revisions following some referee's comments. Ver4: The published version