Some existence theorems of a complete Ricci-flat Kähler metrics on affine algebraic manifolds are established. As an application, we propose a picture describing the behavior of the Ricci-flat Kähler metrics under semi-stable degenerations of polarized K3 surfaces.

This is an expository paper on the moduli space of K3 surfaces with Kähler–Einstein metric. We construct the universal family of Kähler–Einstein metrics on a K3 surface and show that it necessarily includes singular metrics such that some embedded 2-spheres have zero volume. We analyze this degeneration of Kähler–Einstein metrics in detail.

This is an expository paper on the structure of complex surfaces which have the Hirzebruch proportionality $3c_2 = c_1^2$ or $2c_2 = c_1^2$ between their Chern numbers. We characterize surfaces with $3c_2 = c_1^2$ as ball quotients in the category of normal surfaces with branch loci. We discuss the uniformization problem for surfaces with $2c_2 = c_1^2$ from the point of view of Kähler–Einstein metrics and holomorphic conformal structures.