Isotopy of symplectic forms on K3 surfaces to Kähler forms

Determine whether every symplectic form on a K3 surface is isotopic, through symplectic forms, to a Kähler form. Equivalently, prove or refute the conjecture that any symplectic structure on a K3 surface is isotopic to a Kähler one.

Background

The paper proves homological mirror symmetry for projective K3 surfaces under specific conditions and discusses a Kähler upgrade of the symplectic compactification using Gross–Siebert techniques. In this context, the authors note a broad conjecture in symplectic geometry concerning K3 surfaces.

The conjecture asserts that any symplectic form on a K3 surface should be isotopic to a Kähler form. While the authors achieve a Kähler compactification in their specific setting, the general problem remains unresolved in the literature and is highlighted as a notable open question beyond the scope of the present results.