Fano manifolds whose Chern characters satisfy some positivity conditions

TL;DR

This paper studies Fano manifolds with positive Chern characters, showing they have rich structures of rational curves and are covered by higher rational varieties, advancing understanding of their geometric properties.

Contribution

It demonstrates that Fano manifolds with certain positivity conditions on Chern characters admit chains of higher order minimal rational curves and are covered by higher rational varieties.

Findings

Existence of long chains of higher order minimal rational curves.

Fano manifolds with positivity conditions are covered by higher rational varieties.

Structural insights into the geometry of such Fano manifolds.

Abstract

In this paper, we investigate Fano manifolds whose Chern characters satisfy some positivity conditions. We prove that such manifolds admit long chains of higher order minimal families of rational curves and are covered by higher rational varieties.