Global smoothings of degenerate K3 surfaces with triple points

TL;DR

This paper proves the existence of smoothings for certain degenerate K3 surfaces with triple points, extending previous algebraic results to more general complex surfaces using differential geometric methods.

Contribution

It introduces a differential geometric approach to smoothing degenerate K3 surfaces with triple points, broadening the class of surfaces for which smoothings are known to exist.

Findings

Existence of smoothings under suitable conditions

Generalization of Friedman's algebraic results

Applicable to non-Kählerian surfaces

Abstract

Let $X$ be a normal crossing compact complex surface with triple points. We prove that there exists a family of smoothings of $X$ when $X$ satisfies suitable conditions. Since our differential geometric proof also includes the case where $X$ is neither K\"ahlerian nor $H^1(X, \mathcal O_X)=0$, this generalizes Friedman's result on degenerations of $K3$ surfaces in algebraic geometry.