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

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

2006-06-13

arxiv.org/abs/math/0606309v3

Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 51(99) (2008), no. 4, 339--353.

Mathematics

Complex Variables

17 pages, v. 3.0: another error corrected

Scientific paper

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.

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