Minimal complex surfaces with Levi-Civita Ricci-flat metrics

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.

References in corpus (2)

• Inoue surfaces and the Chern-Ricci flow
• Scalar curvature on compact complex manifolds

Cited by in corpus (2)

• Levi-Civita Ricci-flat metrics on compact complex manifolds
• On compact Hermitian manifolds with flat Gauduchon connections