$4$-dimensional almost-K\"ahler Lie algebras of constant Hermitian holomorphic sectional curvature are K\"ahler

TL;DR

This paper proves that all 4-dimensional almost-Kähler Lie algebras with constant Hermitian holomorphic sectional curvature are actually Kähler, establishing a rigidity result in this geometric setting.

Contribution

It demonstrates that under the specified curvature condition, almost-Kähler structures in four dimensions must be Kähler, revealing a rigidity phenomenon.

Findings

All such Lie algebras are Kähler.

The result applies specifically to the canonical Hermitian connection.

It characterizes the geometric structure under constant curvature conditions.

Abstract

We prove that any $4$-dimensional almost-K\"ahler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is K\"ahler.