Higher Fano Manifolds

TL;DR

This paper investigates higher Fano manifolds characterized by positive higher Chern characters, classifies certain rational homogeneous spaces with these properties, and explores conjectural characterizations of projective spaces.

Contribution

It classifies rational homogeneous spaces with positive second and third Chern characters and discusses conjectural characterizations of projective spaces based on higher Fano conditions.

Findings

Only projective spaces and quadrics have positive second and third Chern characters among Picard rank 1 homogeneous spaces.

Classified Fano manifolds of large index with positive higher Chern characters.

Discussed conjectural characterizations of projective spaces and complete intersections.

Abstract

In this paper we address Fano manifolds with positive higher Chern characters. They are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo the Brauer obstruction). Aiming at finding new examples of higher Fano manifolds, we investigate positivity of higher Chern characters of rational homogeneous spaces. We determine which rational homogeneous spaces of Picard rank $1$ have positive second Chern character, and show that the only rational homogeneous spaces of Picard rank $1$ having positive second and third Chern characters are projective spaces and quadric hypersurfaces. We also classify Fano manifolds of large index having positive second and third Chern characters. We conclude by discussing conjectural characterizations of projective spaces and complete intersections in terms of these higher Fano conditions.