Curvature positivity for K\"ahler and quasi-K\"ahler flag manifolds

TL;DR

This paper investigates curvature positivity properties on Kähler and quasi-Kähler flag manifolds, establishing existence results, classifications, and conjectures about the nature of such metrics.

Contribution

It proves that all flag manifolds admit dual-Nakano semi-positive metrics and classifies Kähler flag manifolds with Griffiths semi-positivity, also exploring restrictions and conjectures for quasi-Kähler cases.

Findings

Every flag manifold admits a dual-Nakano semi-positive metric.

Full classification of Kähler flag manifolds with Griffiths semi-positive curvature.

Restrictions and conjectures for quasi-Kähler flag manifolds.

Abstract

In this paper, we study the notions of Griffiths and dual-Nakano positivity for the curvature of the Chern connection on K\"ahler and quasi-K\"ahler flag manifolds, as well as for the complex projective space. In this setting, we prove that every flag manifold endowed with a complex structure admits a metric of dual-Nakano semi-positive curvature, and we give a full classification of K\"ahler flag manifolds with Griffiths semi-positive curvature. Next we prove a series of restrictions for a quasi-K\"ahler flag manifold to have Griffiths semi-positive curvature, and we conjecture that in fact, there are no such metrics for non-integrable almost-complex structures. Lastly, we give a full classification on invariant metrics on the complex projective space with Griffiths and dual-Nakano semi-positive curvature.