Almost toric fibrations on K3 surfaces

TL;DR

This paper demonstrates that generic fibers of K3 surfaces admit almost toric fibrations with natural affine structures, and establishes symplectic Kulikov models for certain hypersurfaces in toric Fano threefolds, connecting to Gross and Siebert's structures.

Contribution

It introduces the existence of almost toric fibrations on K3 surfaces and constructs symplectic Kulikov models for specific hypersurfaces, linking affine structures to known models.

Findings

Generic smooth fibers admit almost toric fibrations.

Smooth anti-canonical hypersurfaces admit symplectic Kulikov models.

Induced affine structures match those in Gross and Siebert's framework.

Abstract

For K\"ahler K3 surfaces we consider Kulikov models of type III tamed by a symplectic form. Our main result shows that the generic smooth fiber admits an almost toric fibration over the intersection complex, which inherits a natural nodal integral affine structure from almost toric fibrations of the boundary divisors. We prove that a smooth anti-canonical hypersurface in a smooth toric Fano threefold, equipped with a toric K\"ahler form, admits a symplectic Kulikov model. Moreover, we demonstrate that the induced integral affine structure on the intersection complex is integral affine isomorphic (up to nodal slides) nodal integral affine structure considered by Gross and Siebert on the boundary of the moment polytope.