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

Abstract

Let XX be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if XX is dd-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 XX, and the first author’s existence result of holomorphic volume forms on global smoothings of XX. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples of dd-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, and K3K3 surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to K3K3 surfaces.

Keywords

• complex surfaces with trivial canonical bundle
• normal crossing varieties
• k3 surfaces
• smoothing
• gluing
• d-semistability
• primary: 58j37
• secondary: 14j28
• 32j15
• 53c56