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

A Hermitian Einstein-Weyl manifold is a complex manifold admitting a\nRicci-flat Kaehler covering W, with the deck transform acting on W by\nhomotheties. If compact, it admits a canonical Vaisman metric, due to\nGauduchon. We show that a Hermitian Einstein-Weyl structure on a compact\ncomplex manifold is determined by its volume form. This result is a conformal\nanalogue of Calabi's theorem stating the uniqueness of Kaehler metrics with a\ngiven volume form in a given Kaehler class. We prove that a solution of a\nconformal version of complex Monge-Ampere equation is unique. We conjecture\nthat a Hermitian Einstein-Weyl structure on a compact complex manifold is\nunique, up to a holomorphic automorphism, and compare this conjecture to\nBando-Mabuchi theorem.\n