Vaisman structures on compact quotients of Lie groups

MARCOS ORIGLIA; ADRIÁN ANDRADA

Workshop; IV School and Workshop on Lie Theory; 2015

The most important class of Hermitian manifolds are the well known K\"ahler manifolds. Another class, much studied,is given by the Locally comformally K\"ahler (LCK) manifolds, that is, an Hermitian manifold whose metric is comformal to a K\"ahler metric in some neighbourhood of each point.Among them, the Vaisman manifolds are very interesting because its topological properties and relations with Sasakian geometry.Our aim is to find Vaisman structures on compact quotients $\Gamma\backslash G$ of a simply connected solvable Lie group $G$ by a lattice $\Gamma$, where these structures come from left-invariant Vaisman structures on $G$, or equivalently, from Vaisman structures on the Lie algebra of $G$.We characterize unimodular solvable Lie algebras with Vaisman structures in terms of K\"ahler flat Lie algebras. Using this characterization we exhibit families of these Lie algebras and Lie groups and we show the existence of lattices on some of these families.