Abstract
In this paper we describe a family of locally conformal Kähler metrics on class 1 Hopf surfaces H α,β containing some recent metrics constructed in [GO98]. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation ℰα,β that is shown to be independent of the metric. We prove with elementary tools that ℰα,β has compact leaves if and only if αm=βn for some integers m and n, namely in the elliptic case. In this case we prove that the leaves of ℰα,β explicitly give the elliptic fibration of H α,β, and we describe the natural orbifold structure on the leaf space.
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Mathematics Subject Classification (2000)
53C55
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Parton, M. Hopf surfaces: locally conformal Kähler metrics and foliations. Ann. Mat. Pura Appl. IV. Ser. 182, 287–306 (2003). https://doi.org/10.1007/s10231-002-0066-9
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DOI: https://doi.org/10.1007/s10231-002-0066-9