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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Locally conformal Kähler structures
in quaternionic geometry


Authors: Liviu Ornea and Paolo Piccinni
Journal: Trans. Amer. Math. Soc. 349 (1997), 641-655
MSC (1991): Primary 53C15, 53C25, 53C55
MathSciNet review: 1348155
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Abstract: We consider compact locally conformal quaternion Kähler manifolds $M$. This structure defines on $M$ a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project $M$ over a quaternion Kähler orbifold $N$ with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler 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 Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of $\mathbb H^n-\{0\}$ by the diagonal action of appropriately chosen discrete subgroups of $CO^+(4)$.


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Additional Information

Liviu Ornea
Affiliation: Faculty of Mathematics, University of Bucharest, 14, Academiei str., 70109 Bucha- rest, Romania
Email: lornea@imar.ro

Paolo Piccinni
Affiliation: Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le A. Moro, 2, I-00185 Roma, Italy
Email: piccinni@axrma.uniromas.it

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01591-2
PII: S 0002-9947(97)01591-2
Keywords: Locally conformal hyperk\"ahler manifold, locally conformal quaternion K\"ahler manifold, Einstein-Weyl structure
Received by editor(s): September 1, 1994
Additional Notes: The first author was supported by C.N.R.\ of Italy, the second author by M.U.R.S.T.\ of Italy and by the E. Schrödinger Institute in Vienna
Article copyright: © Copyright 1997 American Mathematical Society