Balanced and pluriclosed metrics on real semisimple Lie groups

TL;DR

This paper characterizes the existence of balanced and pluriclosed metrics on certain complex manifolds derived from real semisimple Lie groups, revealing they cannot coexist, and revises their classification.

Contribution

It provides a new characterization of metric existence on these manifolds using Vogan diagrams and corrects previous classifications.

Findings

Balanced and pluriclosed metrics cannot coexist on these manifolds.

Existence of such metrics is characterized by Vogan diagrams.

Revised classification of regular complex structures on real semisimple Lie groups.

Abstract

We characterise the existence of balanced and pluriclosed metrics on compact quotients of real semisimple Lie groups equipped with regular complex structures, in terms of Vogan diagrams. Consequently, such complex manifolds cannot simultaneously admit a balanced metric and a pluriclosed metric. Along the way, we revisit and correct the classification of regular complex structures on real semisimple Lie groups.