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

TL;DR

This paper proves the existence of smoothings for certain complex surfaces with trivial canonical bundle and simple normal crossing singularities, using differential geometric methods and explicit local constructions.

Contribution

It introduces a differential geometric approach to smoothability of SNC complex surfaces with trivial canonical bundle, extending algebraic geometry results.

Findings

Existence of smoothings for $d$-semistable SNC surfaces with trivial canonical bundle.

Construction of local smoothings around singularities.

Examples of smoothable surfaces including tori, K3, and Kodaira surfaces.

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.