Minimal Complex Surfaces with Levi-Civita Ricci-flat Metrics

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.