Toric degenerations yielding Lagrangian torus fibrations with controlled singularities

Establish that toric degenerations of Calabi–Yau hypersurfaces and complete intersections admit Lagrangian torus fibrations whose singularities are modeled on a small, finite collection of singularity types, thereby confirming the long-standing conjectural picture underpinning the SYZ program.

Background

The authors highlight a foundational geometric input for the local-to-global SYZ reconstruction program: the existence of Lagrangian torus fibrations for toric degenerations of Calabi–Yau varieties with singularities drawn from a small list of model types. They note that, despite widespread expectations, this has not yet been satisfactorily established.

Confirming this conjecture would provide the requisite differential-geometric scaffolding to build involutive covers by Liouville domains and systematically apply the deformation-theoretic and Mayer–Vietoris tools developed in the paper, thereby enabling explicit mirror reconstructions in broad Calabi–Yau settings.