Complex K3 surfaces containing Levi-flat hypersurfaces

TL;DR

This paper constructs a specific complex K3 surface with a one-parameter family of Levi-flat hypersurfaces, where all leaves are dense, demonstrating a new geometric phenomenon in complex surface theory.

Contribution

It provides the first example of a non-Kummer K3 surface with dense-leaf Levi-flat hypersurfaces, constructed via patching open complex surfaces.

Findings

Existence of a non-Kummer K3 surface with Levi-flat hypersurfaces

Construction method using patching of open complex surfaces

All leaves in the Levi-flat hypersurfaces are dense

Abstract

We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.