Vaisman structures on compacts quotients of Lie groups

MARCOS ORIGLIA; ADRIÁN ANDRADA

Campinas

Workshop; IV School and Workshop on Lie TheoryCampinas; 2015

The most important class of Hermitian manifolds are the well known Kählermanifolds. Another class, much studied,is given by the Locally comformallyKähler (LCK) manifolds, that is, an Hermitian manifold whose metric is com-formal to a Kähler metric in some neighbourhood of each point. Among them,the Vaisman manifolds are very interesting because its topological propertiesand relations with Sasakian geometry. Our aim is to find Vaisman structures oncompact quotients Γ\G of a simply connected solvable Lie group G by a latticeΓ, where these structures come from left-invariant Vaisman structures on G, orequivalently, from Vaisman structures on the Lie algebra of G.We characterize unimodular solvable Lie algebras with Vaisman structures interms of Kähler flat Lie algebras. Using this characterization we exhibit familiesof these Lie algebras and Lie groups and we show the existence of lattices onsome of these families.This is a joint work with Adrián Andrada.