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 Kahler Calabi–Yau surfaces and Hopf surfaces.