[Paper Review] Minimal complex surfaces with Levi-Civita Ricci-flat metrics

Kefeng Liu, Xiaokui Yang|arXiv (Cornell University)|Jun 5, 2017

Geometry and complex manifolds17 references3 citations

TL;DR

This paper classifies minimal complex surfaces admitting Levi-Civita Ricci-flat Hermitian metrics by showing they are either Kähler Calabi-Yau surfaces (Enriques, bi-elliptic, K3, and 2-tori) or Hopf surfaces. It introduces the first Aeppli-Chern class as a bridge between Riemannian and complex geometry and proves that Levi-Civita Ricci-flatness implies vanishing of this class, leading to a classification analogous to the Calabi conjecture in the non-Kähler setting.

ABSTRACT

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli-Chern class on compact complex manifolds, and proved that the $(1,1)$ curvature form of the Levi-Civita connection represents the first Aeppli-Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and classify minimal complex surfaces with Levi-Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi-Civita Ricci-flat metrics are Kähler Calabi-Yau surfaces and Hopf surfaces.

Motivation & Objective

• To establish a geometric link between Riemannian geometry (via the Levi-Civita connection) and complex geometry on non-Kähler Hermitian manifolds.
• To investigate the existence of Levi-Civita Ricci-flat Hermitian metrics on compact complex manifolds, particularly minimal complex surfaces.
• To classify minimal complex surfaces that admit such metrics, extending the Calabi-Yau theorem to the non-Kähler setting.
• To construct explicit examples of Levi-Civita Ricci-flat metrics on diagonal Hopf surfaces, supporting a broader conjecture on their existence.

Proposed method

• Introduces the first Aeppli-Chern class $ c_1^{ ext{AC}}(X) $ as the cohomology class represented by the $(1,1)$-form of the Levi-Civita connection's curvature.
• Uses the identity $ ext{Ric}( ext{LC}) = ext{Ric}( ext{Chern}) - rac{1}{2}(ar heta ar heta^* + heta heta^*) $ to relate the Levi-Civita Ricci curvature to the Chern-Ricci curvature and metric adjoint operators.
• Applies the characterization $ ext{Ric}( ext{LC})( ext{LC}) = 0 $ if and only if $ ext{Ric}( ext{Chern}) = rac{1}{2}(ar heta ar heta^* + heta heta^*) $, where $ heta = ar heta = ext{Ric}( ext{LC}) $, to derive necessary conditions.
• Employs Kodaira-Enriques classification of minimal complex surfaces to analyze possible candidates under the condition $ c_1^{ ext{AC}}(X) = 0 $.
• Constructs a perturbed metric $ ho_g = ho_0 - \frac{1}{n} \sqrt{-1} \partial\bar\partial \log|z|^2 $ on diagonal Hopf manifolds to explicitly realize Levi-Civita Ricci-flatness.
• Uses Gauduchon metric techniques and curvature integration to rule out Levi-Civita Ricci-flatness on Inoue and Kodaira surfaces.

Experimental results

Research questions

• RQ1 Which minimal complex surfaces admit a Hermitian metric for which the Levi-Civita connection has Ricci curvature zero?
• RQ2 Does the vanishing of the first Aeppli-Chern class $ c_1^{ ext{AC}}(X) $ imply the existence of a Levi-Civita Ricci-flat Hermitian metric?
• RQ3 Can Levi-Civita Ricci-flat metrics be constructed on Hopf surfaces, and do they exist on all such surfaces?
• RQ4 Why do Inoue and Kodaira surfaces fail to support Levi-Civita Ricci-flat metrics despite having $ c_1(X) = c_1^{ ext{BC}}(X) = c_1^{ ext{AC}}(X) = 0 $?
• RQ5 How does the Levi-Civita Ricci curvature differ from the Chern-Ricci curvature in non-Kähler geometry, and what equations govern it?

Key findings

• Minimal complex surfaces admitting Levi-Civita Ricci-flat Hermitian metrics are precisely Kähler Calabi-Yau surfaces (Enriques, bi-elliptic, K3, and 2-tori) and Hopf surfaces.
• The first Aeppli-Chern class $ c_1^{ ext{AC}}(X) $ is represented by the $(1,1)$-form of the Levi-Civita connection's curvature, providing a natural link between Riemannian and complex geometry.
• For diagonal Hopf surfaces, the metric $ \omega_g = \omega_0 - \frac{1}{n}\sqrt{-1}\partial\bar\partial\log|z|^2 $ is explicitly shown to be Levi-Civita Ricci-flat.
• The Levi-Civita Ricci-flat condition $ \mathfrak{Ric}(\omega) = 0 $ is equivalent to $ \text{Ric}(\omega) = \frac{1}{2}(\partial\partial^*\omega + \bar\partial\bar\partial^*\omega) $, a non-elliptic PDE system.
• Inoue and Kodaira surfaces do not admit Levi-Civita Ricci-flat metrics because any Gauduchon metric on them has negative total Ricci curvature integral, contradicting the required identity.
• The paper conjectures that all Hopf surfaces support Levi-Civita Ricci-flat metrics, though only diagonal ones are constructed explicitly.