Vaisman solvmanifolds with trivial canonical bundle

ADRIÁN ANDRADA; MARCOS ORIGLIA

Brasilia

Workshop; VI School and Workshop on Lie Theory; 2019

A Hermitian manifold (M, J, g) is called locally conformal Kähler (LCK) if each point of M has a neighborhood where the metric g is conformal to a Kähler metric with respect to J. Equivalently, there exists a closed 1-form θ on M such that dω = θ ∧ ω, where ω denotes the fundamental 2-form associated to (J, g), defined by ω(·, ·) = g(J ·, ·). The 1-form θ is called the Lee form.If the Lee form is parallel with respect to the Levi-Civita connection associated to g, the LCK structure is called Vaisman. The family of Vaisman manifolds is very important, due to its relations with Sasakian geometry, and they have been deeply studied since the seminal work of I. Vaisman in the ?80s.In this talk we will review invariant LCK structures on solvmanifolds, i.e., compact quotients of a simply connected solvable Lie group G by a discrete subgroup. We will characterize the Lie algebras associated to a solvmanifold admitting a Vaisman structure, and using this characterization we will exhibit many examples of Vaisman solvmanifolds whose underlying complex manifolds have trivial canonical bundle.