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

TL;DR

This paper explores the uniqueness of Hermitian Einstein-Weyl structures on compact complex manifolds, establishing a conformal analogue of Calabi's theorem and proving the uniqueness of solutions to a conformal Monge-Ampère equation.

Contribution

It introduces a conformal version of Calabi's theorem for Hermitian Einstein-Weyl structures and proves the uniqueness of solutions to the related Monge-Ampère equation.

Findings

Hermitian Einstein-Weyl structures are determined by their volume form.

Solutions to the conformal Monge-Ampère equation are unique.

A conjecture on the uniqueness of these structures up to automorphisms is proposed.

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.