Reduction of Vaisman structures in complex and quaternionic geometry

Abstract

We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kaehler manifold (K,Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover we define locally conformal hyperkaehler reduction as an equivariant version of hyperkaehler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally we show that locally conformal hyperkaehler reduction induces hyperkaehler 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 hyperkaehler reduction.

29 pages; Section 4 changed (and accordingly the Introduction); Remark 8.2 added; References updated

References in corpus (3)

• Reduction of HKT-Structures
• Locally conformal Kaehler reduction
• Potential one-forms for hyperkähler structures with torsion

Cited by in corpus (14)

• Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds
• Hamiltonian symmetries and reduction in generalized geometry
• LCK rank of locally conformally Kahler manifolds with potential
• Locally conformally Kähler manifolds. A selection of results
• Examples of non-trivial rank in locally conformal Kähler geometry
• Locally conformally Kähler structures on unimodular Lie groups
• Toric Vaisman Manifolds
• Locally conformally symplectic convexity
• Cohomologies of locally conformally symplectic manifolds and solvmanifolds
• Locally conformally Kähler solvmanifolds: a survey
• CR-submanifolds. A class of examples
• LCK metrics on toric LCS manifolds
• Hermitian ranks of compact complex manifolds
• Holomorphic submersions of locally conformally Kähler manifolds