Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle

Abstract

Let $X$ be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if $X$ is $d$-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of $X$, and the first author's existence result of holomorphic volume forms on global smoothings of $X$. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples of $d$-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces and $K3$ surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to $K3$ surfaces.