On SYZ mirrors of Hirzebruch surfaces

Abstract

The Strominger-Yau-Zaslow (SYZ) approach to mirror symmetry constructs a mirror space and a superpotential from the data of a Lagrangian torus fibration on a K\"ahler manifold with effective first Chern class. For K\"ahler manifolds whose first Chern class is not nef, the SYZ construction is further complicated by the presence of additional holomorphic discs with non-positive Maslov index. In this paper, we study SYZ mirror symmetry for two of the simplest non-Fano toric examples: the Hirzebruch surfaces F_3 and F_4. Our approach is to regularize moduli spaces of stable holomorphic discs using obstruction sections arising from infinitesimal deformations of the complex structure. For F_3, we determine the SYZ mirror associated to generic regularizing perturbations of the complex structure, and demonstrate that the mirror depends on the choice of perturbation. For F_4, we determine the SYZ mirror for a specific regularizing perturbation, where the mirror superpotential is an explicit infinite Laurent series. Finally, we relate this superpotential to those arising from other perturbations of F_4, as determined in the literature \cite{CPS24, BGL25}, via a scattering diagram.