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Abstract. A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering , with monodromy acting on by Kähler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on . We prove that any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold (Cn\0)Z. As an application, we obtain that any Sasakian manifold has a contact immersion to an odd-dimensional sphere.

We define reduction of locally conformal Kähler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We give two independent, equivalent definitions, the first via local charts, the second via lifting to Kähler reduction of the universal covering. By a recent result of Kamishima and the second author, in the Vaisman case (that is, when a metric in the conformal class has parallel Lee form) if the manifold is compact its universal covering comes equipped with the structure of Kähler cone over a Sasaki compact manifold. We show the compatibility between our reduction and Sasaki reduction, hence describing a subgroup of automorphisms whose action causes the quotient to bear a Vaisman structure. Then we apply this theory to construct a wide class of Vaisman manifolds.

It is shown that an HKT space with closed parallel potential 1-form has D(2, 1; −1) symmetry. Every locally conformally hyper-Kähler manifold generates this type of geometry. The HKT spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic condition on their torsion.

Let (M, g, H) be a quaternion Hermitian manifold. The additional datum of a torsion-free connection D preserving both the quaternionic structure H and the conformal class of g defines on M the structure of quaternion Hermitian–Weyl manifold. Under the compactness assumption of both M and the leaves of a canonical foliation, M is here shown to project on a locally 3-Sasakian orbifold P. Then M is proved to admit both a compatible global complex structure and a finite covering M carrying a hyperhermitian–Weyl structure. The uniqueness of the Weyl structure compatible with a given quaternion Hermitian metric and some restrictions on the Betti numbers are also obtained.