  A Hermitian Einstein-Weyl manifold is a complex manifold admitting a
Ricci-flat Kaehler covering W, with the deck transform acting on W 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 Kaehler metrics with a
given volume form in a given Kaehler class. We prove that a solution of a
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.
