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 $1$, $(-K_X)^3\leq 72$. 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.