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 toric examples: the non-Fano Hirzebruch surfaces F_3 and F_4. For F_3, we determine the SYZ mirror associated to generic perturbations of the complex structure, and demonstrate that the SYZ mirror depends on the choice of perturbation. For F_4, we determine the SYZ mirror for a specific perturbation of complex structure, where the mirror superpotential is an explicit infinite Laurent series. Finally, we relate this superpotential to those arising from other perturbations of F_4 via a scattering diagram.