On Lie groups with left-invariant symplectic or Kählerian structures
In this paper the authors search for conditions, on a connected Lie group, which ensure the existence of invariant symplectic or Kählerian structures. Every Kählerian manifold has a canonical symplectic structure but the converse does not hold in general. There has been recent interest in examples of compact symplectic manifolds which do not admit Kählerian structures. The main results of the present paper are: Theorem 5. A simply connected Kählerian group is a direct product of a unimodular Kählerian group which is metabelian and flat by a Kählerian group with non vanishing Ricci curvature which either admits a holomorphic fibration with flat Kählerian fibers of admits an exact symplectic structure given by its Ricci 2-form. Theorem 6. Let G be a symplectic group admitting a left invariant metric such that its Ricci curvature is \(\geq 0\). The group G is necessarily unimodular, metabelian and admits a left invariant Kählerian structure which is flat.
scientific article; zbMATH DE number 4088448