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

Abstract

Abstract Let X X be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if X X is d 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 X , and the first author’s existence result of holomorphic volume forms on global smoothings of X 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 d -semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, and K 3 K3 surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to K 3 K3 surfaces.