[Paper Review] Levi-Civita Ricci-flat metrics on compact complex manifolds

Jie He, Kefeng Liu+1 more|arXiv (Cornell University)|Jun 18, 2018

Geometry and complex manifolds18 references3 citations

TL;DR

This paper classifies compact complex surfaces admitting Levi-Civita Ricci-flat Hermitian metrics, proving they are either Kähler Calabi-Yau surfaces (Enriques, bi-elliptic, K3, or 2-tori) or Hopf surfaces. It constructs explicit Levi-Civita Ricci-flat metrics on Hopf surfaces of type 1 using conformal methods and functional analysis, demonstrating the existence of non-Kähler Calabi-Yau metrics via the Levi-Civita connection, a novel approach distinct from Chern Ricci-flatness.

ABSTRACT

In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces or Hopf surfaces.

Motivation & Objective

• To classify compact complex surfaces that admit Levi-Civita Ricci-flat Hermitian metrics.
• To investigate the geometric distinction between Levi-Civita Ricci-flatness and Chern Ricci-flatness in non-Kähler complex geometry.
• To construct explicit examples of Levi-Civita Ricci-flat metrics on non-Kähler manifolds, particularly Hopf surfaces.
• To clarify the role of the first Aeppli-Chern class and Levi-Civita Ricci curvature in non-Kähler Calabi-Yau geometry.

Proposed method

• Uses the Levi-Civita connection on the holomorphic tangent bundle to define the Levi-Civita Ricci curvature as a representative of the first Aeppli-Chern class.
• Applies conformal methods and functional analysis to solve the non-elliptic, non-Monge-Ampère equation for Levi-Civita Ricci-flatness: $\mathfrak{Ric}(\omega) = \frac{1}{2}(\partial\partial^*\omega + \overline{\partial}\overline{\partial}^*\omega)$.
• Constructs a one-parameter family of Hermitian metrics $\omega_\lambda$ on Hopf surfaces via a conformal factor $\Phi$ and a parameter $\lambda$, showing $\partial^*\omega_\lambda = \frac{\sqrt{-1}}{1+\lambda}\overline{\partial}\log\Phi$.
• Derives the Levi-Civita Ricci curvature formula $\mathfrak{Ric}(\omega_\lambda) = \left(2 - \frac{1}{1+\lambda}\right)\sqrt{-1}\partial\overline{\partial}\log\Phi + 3\sqrt{-1}\partial\overline{\partial}\log\Delta$.
• Selects $\lambda = -\frac{1}{2}$ to eliminate the $\log\Phi$ term, yielding $\mathfrak{Ric}(\omega_{-1/2}) = 3\sqrt{-1}\partial\overline{\partial}\log\Delta$, and rescales by $\Delta^3$ to achieve $\mathfrak{Ric}(\omega) = 0$.
• Employs explicit matrix computations for metric and inverse metric components, and uses Christoffel symbols to compute $\partial^*$ and $\overline{\partial}^*$.

Experimental results

Research questions

• RQ1 Which compact complex surfaces admit Levi-Civita Ricci-flat Hermitian metrics?
• RQ2 How does Levi-Civita Ricci-flatness differ from Chern Ricci-flatness in non-Kähler geometry?
• RQ3 Can explicit Levi-Civita Ricci-flat metrics be constructed on non-Kähler Calabi-Yau manifolds like Hopf surfaces?
• RQ4 Why do Kodaira surfaces fail to admit Levi-Civita Ricci-flat metrics despite having trivial first Chern and Aeppli-Chern classes?
• RQ5 Do all Hopf surfaces support Levi-Civita Ricci-flat metrics, or only those of type 1?

Key findings

• Compact complex surfaces with Levi-Civita Ricci-flat metrics are minimal and fall into five classes: Enriques, bi-elliptic, K3, 2-tori, and Hopf surfaces.
• Kähler Calabi-Yau surfaces (Enriques, bi-elliptic, K3, 2-tori) admit Levi-Civita Ricci-flat metrics, consistent with Yau’s theorem.
• Hopf surfaces of type 1 admit explicit Levi-Civita Ricci-flat metrics constructed via conformal deformation of a Hermitian metric using a function $\Phi$ and parameter $\lambda = -1/2$.
• The constructed metric $\omega = \Delta^3 \omega_{-1/2}$ satisfies $\mathfrak{Ric}(\omega) = 0$, confirming Levi-Civita Ricci-flatness.
• Kodaira surfaces do not admit Levi-Civita Ricci-flat metrics despite having $c_1^{\mathrm{BC}} = c_1^{\mathrm{AC}} = 0$, highlighting a key distinction from Chern Ricci-flatness.
• The paper conjectures that all Hopf surfaces support Levi-Civita Ricci-flat metrics, though only type 1 is constructed explicitly.