Abelian Hermitian geometry
arXiv:1106.6268
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 Kähler 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.
14 pages