Liviu Ornea, Paolo Piccinni
Locally Conformal Kaehler Structures in Quaternionic Geometry
The paper is published: Trans. Am. Math. Soc. 349, 2 (1997) 641-655

MSC:

53C15 General geometric structures on manifolds (almost complex, contact, symplectic, almost product structures, etc.)

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

53C55 Hermitian and Kahlerian manifolds, See also {32Cxx}

Abstract: We consider compact {\it locally conformal quaternion K\"ahler} manifolds
$M$. This structure defines on $M$ a canonical foliation, which we assume to
have compact leaves. We prove that the local quaternion
K\"ahler metrics are Ricci-flat and allow to project $M$ over a quaternion
K\"ahler orbifold
$N$ with fibers conformally flat 4-dimensional real Hopf manifolds. This
fibration was known for the subclass of {\it locally conformal hyperk\"ahler}
manifolds; in this case we make some observations on the fibers' structure and
obtain restrictions on the Betti numbers. In the homogeneous case
$N$ is shown to be a manifold and this allows a classification . Examples of
locally conformal quaternion K\"ahler manifolds (some with a global complex
structure, some locally conformal hyperk\"ahler) are the Hopf manifolds
quotients of $\Bbb H^n - \{0\}$ by the diagonal action of appropriately chosen
discrete subgroups of $CO^+(4)$.

Keywords: locally conformal hyperkaehler manifold, locally conformal quaternion Kaehler manifold, Einstein-Weyl structure