Polarized minimal families of rational curves and higher Fano manifolds

In this paper we investigate Fano manifolds $X$ whose Chern characters ${\rm ch}_k(X)$ satisfy some positivity conditions. Our approach is via the study of {\it polarized minimal families of rational curves} $(H_x,L_x)$ through a general point $x\in X$. First we translate positivity properties of the Chern characters of $X$ into properties of the pair $(H_x,L_x)$. This allows us to classify polarized minimal families of rational curves associated to Fano manifolds $X$ satisfying ${\rm ch}_2(X) \geq 0$ and ${\rm ch}_3(X) \geq 0$. This classification enables us to find new examples of higher Fano manifolds. We also provide sufficient conditions for these manifolds to be covered by subvarieties isomorphic to ${\Bbb P}^2$ and ${\Bbb P}^3$.