Polarized minimal families of rational curves and higher Fano manifolds

Abstract

In this paper we investigate Fano manifolds $X$ whose Chern characters $c{h}_k(X)$ satisfy some positivity conditions. Our approach is via the study of 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 $c{h}_2(X)\ge 0$ and $c{h}_3(X)\ge 0$. As a first application, we provide sufficient conditions for these manifolds to be covered by subvarieties isomorphic to ${\mathbb{P}}^2$ and ${\mathbb{P}}^3$. Moreover, this classification enables us to find new examples of Fano manifolds satisfying $c{h}_2(X)\ge 0$.

17 pages