Hopf surfaces: a family of locally conformal Kaehler metrics and elliptic fibrations

We describe a family of locally conformal Kaehler metrics on class 1 Hopf surfaces H containing some recent metrics constructed by P. Gauduchon and L. ornea. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation E that is shown to be independent of the metric. We elementary prove that E has compact leaves if and only if H is elliptic. In this case the leaves of E give explicitly the elliptic fibration of H, and the natural orbifold structure on the leaf space is illustrated.