Twisted K\"ahler-Einstein metrics on flag varieties

TL;DR

This paper characterizes invariant twisted K"ahler-Einstein metrics on flag varieties, explores their applications to constant scalar curvature metrics, and provides bounds on Ricci curvature using Lie theory tools.

Contribution

It offers a detailed description of invariant twisted K"ahler-Einstein metrics on flag varieties and connects these to scalar curvature and Ricci bounds.

Findings

Explicit description of tKE metrics on various flag varieties

Inequalities for volume bounds derived from Lie theory

Applications to existence of constant scalar curvature K"ahler metrics

Abstract

In this paper, we describe invariant twisted K\"ahler-Einstein (tKE) metrics on flag varieties. We also explore some applications of the ideas involved in the proof of our main result to the existence of invariant twisted constant scalar curvature K\"{a}hler metrics. Also, we provide a precise description for the greatest Ricci lower bound of an arbitrary K\"{a}hler class on a flag variety. By means of this description, we establish some inequalities related to optimal volume upper bounds for K\"{a}hler metrics just using tools from Lie theory. Further, we describe the set of tKE metrics for several examples, including full flag varieties, the projectivization of the tangent bundle of $\mathbb{P}^{n+1}$, and families of flag varieties with Picard number $2$.