[Paper Review] Einstein-Weyl structures on complex manifolds and conformal version of Monge-Ampere equation

Liviu Ornea, Misha Verbitsky|arXiv (Cornell University)|Jun 13, 2006

Geometry and complex manifolds19 references3 citations

TL;DR

This paper establishes the uniqueness of Hermitian Einstein-Weyl structures on compact complex manifolds up to holomorphic automorphisms, by proving that such structures are uniquely determined by their volume form and Lee class. It introduces a conformal analogue of the complex Monge-Ampère equation and shows its solution is unique, generalizing Calabi’s theorem to locally conformally Kähler geometry.

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.

Motivation & Objective

• To establish the uniqueness of Hermitian Einstein-Weyl structures on compact complex manifolds, analogous to Calabi’s uniqueness result for Kähler-Einstein metrics.
• To formulate and prove the uniqueness of solutions to a conformal version of the complex Monge-Ampère equation on locally conformally Kähler (LCK) manifolds.
• To investigate the conformal analogue of the Bando-Mabuchi theorem in the context of Einstein-Weyl geometry.
• To explore the relationship between Einstein-Weyl structures, Vaisman manifolds, and Sasakian geometry via the Kähler covering of the manifold.
• To conjecture that Einstein-Weyl structures on compact complex manifolds are unique up to holomorphic automorphisms, generalizing the Kähler-Einstein uniqueness result.

Proposed method

• Define a Hermitian Einstein-Weyl manifold as a complex manifold admitting a Ricci-flat Kähler covering where the deck transformations act by holomorphic homotheties.
• Use the Gauduchon metric as the canonical representative in the conformal class of metrics on a compact LCK-manifold.
• Formulate the conformal Monge-Ampère equation as finding a Gauduchon metric with a prescribed volume form, given a fixed Lee class.
• Prove uniqueness of the solution to this conformal Monge-Ampère equation using cohomological arguments and properties of the weight bundle $L_{\mathbb{C}}$.
• Relate the Einstein-Weyl condition to the canonical bundle via the isomorphism $L_{\mathbb{C}}^{n} \cong K^{-1}$ on the Kähler covering.
• Analyze the transversal Monge-Ampère equation on the covering space, showing that the condition $\widetilde{\omega}_1^n = \lambda \widetilde{\omega}_2^n$ leads to a conformal version of the Aubin-Calabi-Yau equation with $\varepsilon = 1$.

Experimental results

Research questions

• RQ1 Is the solution to the conformal Monge-Ampère equation on a compact Vaisman manifold unique for a given volume form and Lee class?
• RQ2 Can the uniqueness of Einstein-Weyl structures on compact complex manifolds be established, up to holomorphic automorphisms, analogous to the Bando-Mabuchi theorem in Kähler geometry?
• RQ3 How does the canonical class of the Kähler covering relate to the weight bundle in the context of Einstein-Weyl structures?
• RQ4 To what extent does the Calabi-Yau theorem generalize to the conformal setting of locally conformally Kähler manifolds?
• RQ5 What is the precise relationship between Einstein-Weyl structures on a manifold and Kähler-Einstein metrics on its associated Fano base?

Key findings

• A Hermitian Einstein-Weyl structure on a compact complex manifold is uniquely determined by its volume form and Lee class, generalizing Calabi’s uniqueness theorem to the conformal setting.
• The solution to the conformal Monge-Ampère equation on a compact Vaisman manifold is unique, provided the Lee class and volume form are fixed.
• The Kähler covering $\widetilde{M}$ of a Hermitian Einstein-Weyl manifold is Ricci-flat and admits a nowhere-vanishing holomorphic volume form that is equivariant under the monodromy action.
• The isomorphism $L_{\mathbb{C}}^{n} \cong K^{-1}$ holds for Einstein-Weyl LCK-manifolds, where $L_{\mathbb{C}}$ is the complexified weight bundle and $K$ the canonical bundle.
• The Einstein-Weyl condition on the covering space leads to a transversal Monge-Ampère equation of the form $\log\left(\frac{\det(\eta - dd^c\psi)}{\det\eta}\right) = \psi + \text{const}$, with $\varepsilon = 1$, which is the conformal analogue of the Aubin-Calabi-Yau equation.
• The conjecture that Einstein-Weyl structures are unique up to holomorphic automorphisms implies the Bando-Mabuchi theorem, as shown by constructing a correspondence between Einstein-Weyl structures on certain manifolds and Kähler-Einstein metrics on Fano manifolds.