Abelian complex structures and related geometries

A. ANDRADA, M.L. BARBERIS, I. DOTTI

Marburg

Workshop; Workshop on geometric structures on manifolds and their applications; 2012

Philipps Universität Marburg

We describe the structure of Lie groups admitting abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian metric, we prove that such a metric is Kähler if and only if the Lie group is the direct product of several copies of the hyperbolic plane by a Euclidan factor. On the other hand, if a Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, we show that the Lie group is abelian.