EinsteinWeyl structures on complex manifolds and conformal version of MongeAmpere equation
Abstract
A Hermitian EinsteinWeyl manifold is a complex manifold admitting a Ricciflat 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 EinsteinWeyl 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 MongeAmpere equation is unique. We conjecture that a Hermitian EinsteinWeyl structure on a compact complex manifold is unique, up to a holomorphic automorphism, and compare this conjecture to BandoMabuchi theorem.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606309
 Bibcode:
 2006math......6309O
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Algebraic Geometry;
 Mathematics  Differential Geometry
 EPrint:
 17 pages, v. 3.0: another error corrected