Global stability of the Pluriclosed flow on compact simply-connected simple Lie groups of rank two

TL;DR

This paper investigates the stability of the pluriclosed flow on compact simply-connected simple Lie groups of rank two by analyzing their cohomology, showing that certain cohomologies are generated by invariant forms and computing specific cohomology structures.

Contribution

It provides the first computation of (1,1)-Aeppli cohomology for these groups and establishes stability results for the pluriclosed flow on them.

Findings

Aeppli cohomology is one-dimensional and generated by Bismut flat metrics.

Dolbeault, Bott-Chern, and Aeppli cohomologies are generated by invariant forms.

Computed Bott-Chern diamonds for SU(3) and Spin(5) with invariant complex structures.

Abstract

We compute the (1,1)-Aeppli cohomology of compact simply-connected simple Lie groups of rank two. In particular, we verify that they are of dimension one and generated by the classes of the Bismut flat metrics coming from the Killing forms. This yields a result on the stability of the pluriclosed flow on these manifolds. Moreover, we show that for compact simply-connected simple Lie groups of rank two the Dolbeaut cohomology, as well as the Bott-Chern and the Aeppli cohomologies, arise from just the left-invariant forms and we computed the whole Bott-Chern diamonds of SU(3) and Spin(5) when they are equipped with a left-invariant isotropic complex structure.