Caltech/USC Joint Algebra & Geometry Seminar (1/2)

The SYZ approach to mirror symmetry for Fano toric varieties is well understood, with the mirror superpotential governed by Maslov index two holomorphic discs. For non-Fano manifolds, however, additional discs of non-positive Maslov index appear and fundamentally alter the SYZ construction. In this talk, I will use the non-Fano Hirzebruch surfaces F_3 and F_4 to illustrate that the resulting mirror can depend on the choice of perturbation used to regularize disc moduli spaces. For F_4, I will present a perturbation for which the mirror superpotential becomes an explicit infinite Laurent series, and explain how it is related to other superpotentials via a scattering diagram.