Second Chern class of Fano manifolds and anti-canonical geometry

Let $X$ be a Fano manifold of Picard number one. We establish a lower bound for the second Chern class of $X$ in terms of its index and degree. As an application, if $Y$ is a $n$-dimensional Fano manifold with $-K_Y=(n-3)H$ for some ample divisor $H$, we prove that $h^0(Y,H)\ge n-2$. Moreover, we show that the rational map defined by $\mathrm{\mid }mH\mathrm{\mid }$ is birational for $m\ge 5$, and the linear system $\mathrm{\mid }mH\mathrm{\mid }$ is basepoint free for $m\ge 7$. As a by-product, the pluri-anti-canonical systems of singular weak Fano varieties of dimension at most $4$ are also investigated.