Reduction of Vaisman structures in complex and quaternionic geometry

[摘要] We consider locally conformal Kahler geometry as an equivariant (homothetic) Kahler geometry: a locally conformal Kahler manifold is, up to equivalence, a pair (K, Gamma), where K is a Kohler manifold and Gamma is a discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kahler manifold (K, Gamma) as the rank of a natural quotient of Gamma, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kahler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover, we define locally conformal hyperKahler reduction as an equivariant version of hyperKahler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally, we show that locally conformal hyperKahler reduction induces hyperKahler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperKahler reduction. (c) 2006 Elsevier B.V. All rights reserved.

[发布日期] 2006-12-01

[关键词] Hamiltonian action;Hopf manifold;hypercomplex manifold;HKT manifold;Lee form;locally conformal Kahler manifold;Sasakian manifold;3-sasakian manifold;symplectic reduction;Vaisman manifold