A Kawamata--Miyaoka type inequality for Fano varieties of arbitrary Picard number

Abstract

Let $X$ be a $\mathbb Q$-factorial canonical weak Fano variety of dimension $n\geq 2$. We show that if the $\mathbb Q$-Fano index $q_{\mathbb Q}(X)\geq 3$, then $X$ satisfies a Kawamata--Miyaoka type inequality: \[c_1(X)^n\leq 4\,\hat c_2(X)\cdot c_1(X)^{n-2}.\] As an application, we show that the $\mathbb Q$-Fano index of a Gorenstein canonical Fano $3$-fold lies in the set $\{m\in\mathbb Z_{>0}\mid m\leq 22\} \cup\{24,30,42\}$.