Abelian Hermitian geometry

TL;DR

This paper investigates Lie groups with abelian complex and Hermitian structures, characterizing when such structures are Kähler and linking flat canonical connections to abelian Lie groups.

Contribution

It provides a classification of Hermitian structures on Lie groups with abelian complex structures, identifying conditions for Kählerness and flatness of canonical connections.

Findings

Hermitian structure is Kähler iff the Lie group is a product of hyperbolic planes and Euclidean spaces.

Flat first canonical connection implies the Lie group is abelian.

Characterization of abelian Lie groups with invariant complex and Hermitian structures.

Abstract

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is K\"ahler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.