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 -KY=(n-3)H for some ample divisor H, we prove that h0(Y,H)>= n-2. Moreover, we show that the rational map defined by vertical bar mH vertical bar is birational for m >= 5, and the linear system vertical bar mH vertical bar is basepoint free for m >= 7. As a by-product, the pluri-anti-canonical systems of singular weak Fano varieties of dimension at most 4 are also investigated.