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

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