Polarized minimal families of rational curves and higher Fano manifolds

TL;DR

This paper classifies certain Fano manifolds with positive Chern characters by studying polarized minimal families of rational curves, leading to new examples and geometric insights.

Contribution

It introduces a classification of polarized minimal families of rational curves on Fano manifolds with positive Chern characters, linking positivity to geometric structures.

Findings

Classified polarized minimal families of rational curves for Fano manifolds with $ch_2 \,\geq 0$ and $ch_3 \,\geq 0$

Provided conditions under which these manifolds are covered by projective spaces

Discovered new examples of Fano manifolds with nonnegative Chern characters

Abstract

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 $ch_2(X)\geq0$.