Einstein-Weyl structures on complex manifolds and conformal version of Monge-Ampere equation

A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kahler covering (M) over bar, with the deck group acting on (M) over bar by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its volume form. This result is a conformal analogue of Calabi's theorem stating the uniqueness of Kahler metrics with a given volume form in a given Kahler class. We prove that, the solution of the conformal version of complex Monge-Ampere equation is unique. We conjecture that a Hermitian Einstein-Weyl structure on a compact complex manifold is unique, up to a holomorphic automorphism, and compare this conjecture to Bando-Mabuchi theorem.