Locally conformally Kahler metrics on Hopf surfaces

A primary Hopf surface is a compact complex surface with universal cover C- 2 - {(0,0)} and cyclic fundamental group generated by the transformation (u ,v) bar right arrow (alpha u + lambda v(m), beta v), m is an element of Z, and alpha, beta, gamma is an element of C such that \ alpha \ greater than or equal to \ beta \ > 1 and (alpha - beta(m))lambda = 0. Being diffeomorph ic with S-3 x S-1 Hopf surfaces cannot admit any Kahler metric. However, it was known that for lambda = 0 and \ alpha \ = \ beta \ they admit a locall y conformally Kahler metric with parallel Lee form. We here provide the con struction of a locally conformally Kahler metric with parallel Lee form for all primary Hopf surfaces of class 1 (lambda = 0). We also show that these metrics are obtained via a Riemannian suspension over S-1, by deforming th e canonical Sasakian structure of S-3 by a Hermitian quadratic form of C-2. We finally infer the existence of a locally conformally Kahler metric for all primary Hopf surfaces by a deformation argument due to C. LeBrun.