Kawamata-Miyaoka-type inequality for $\mathbb Q$-Fano varieties with canonical singularities II: Terminal $\mathbb Q$-Fano threefolds

Abstract

We prove an optimal Kawamata-Miyaoka-type inequality for terminal $\mathbb Q$-Fano threefolds with Fano index at least $3$. As an application, any terminal $\mathbb Q$-Fano threefold $X$ satisfies the following Kawamata-Miyaoka-type inequality \[ c_1(X)^3 < 3c_2(X)c_1(X). \]