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

Let $X$ be a $\mathbb{Q}$-factorial canonical weak Fano variety of dimension $n\ge 2$. We show that if the $\mathbb{Q}$-Fano index $q_{\mathbb{Q}}(X)\ge 3$, then $X$ satisfies a Kawamata–Miyaoka type inequality: $$c_1(X{)}^n\le 4{\widehat{c}}_2(X)\cdot c_1(X{)}^{n-2}\mathrm{.}$$ As an application, we show that the $\mathbb{Q}$-Fano index of a Gorenstein canonical Fano $3$-fold lies in the set $\{m\in {\mathbb{Z}}_{>0}\mid m\le 22\}\cup \{24,30,42\}$.