Einstein-Weyl structures on complex manifolds and conformal version of Monge-Ampere equation
Abstract
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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- June 2006
- DOI:
- 10.48550/arXiv.math/0606309
- arXiv:
- arXiv:math/0606309
- Bibcode:
- 2006math......6309O
- Keywords:
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- Mathematics - Complex Variables;
- Mathematics - Algebraic Geometry;
- Mathematics - Differential Geometry
- E-Print:
- 17 pages, v. 3.0: another error corrected