Polarized minimal families of rational curves and higher Fano manifolds

Abstract. In this paper we investigate Fano manifolds X whose Chern char-acters chk(X) satisfy some positivity conditions. Our approach is via the study of polarized minimal families of rational curves (Hx, Lx) through a general point x ∈ X. First we translate positivity properties of the Chern characters of X into properties of the pair (Hx, Lx). This allows us to classify polarized minimal families of rational curves associated to Fano manifolds X satisfying ch2(X) ≥ 0 and ch3(X) ≥ 0. As a first application, we provide sufficient conditions for these manifolds to be covered by subvarieties isomorphic to P2 and P3. Moreover, this classification enables us to find new examples of Fano manifolds satisfying ch2(X) ≥ 0. 1