Polarized minimal families of rational curves and higher Fano manifolds

Details Polarized minimal families of rational curves and higher Fano manifolds Polarized minimal families of rational curves and higher Fano manifolds

2009-06-30

arxiv.org/abs/0906.5388v1

Mathematics

Algebraic Geometry

17 pages

Scientific paper

In this paper we investigate Fano manifolds $X$ whose Chern characters $ch_k(X)$ 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 $ch_2(X)\geq0$ and $ch_3(X)\geq0$. 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 $ch_2(X)\geq0$.

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