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A locally conformally Kähler (l.c.K.) manifold is a complex manifold admitting a Kähler covering˜M , with each deck transformation acting by Kähler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties oñ M. We prove a structure theorem for compact Vaisman manifolds. Every compact Vaisman(More)
A nilmanifold is a quotient of a nilpotent group G by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a G-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of G-invariant complex structures which(More)
We construct examples of compact hyperkähler manifolds with torsion (HKT manifolds) which are not homogeneous and not locally conformal hyperkähler. Consider a total space T of a tangent bundle over a hyperkähler manifold M. The manifold T is hypercomplex, but it is never hy-perkähler, unless M is flat. We show that T admits an HKT-structure. We also prove(More)
A hypercomplex manifold is a manifold equipped with a triple of complex structures I, J, K satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hy-percomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics, and prove a quaternionic analogue of(More)
A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering˜M , with monodromy acting oñ M by Kähler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties oñ M. We prove that any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold (C n(More)
Let M be a compact Riemannian manifold equipped with a parallel differential form ω. We prove a version of Kähler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquo-tient (H * c (M), d), called the pseudocohomology of M. When M is compact and Kähler and ω is its Kähler form, (H * c (M), d) is(More)