Mirror symmetry for lattice polarized del Pezzo surfaces

Mirror symmetry for lattice polarized K3 surfaces is a well-established correspondence. I will discuss the possibility of extending this theory to log Calabi–Yau surfaces, and in particular to weak del Pezzo surfaces. I will propose a way to endow a weak del Pezzo surface with a lattice polarization and suggest how one might define the mirror notion of a lattice polarized Landau-Ginzburg model. Finally, I will discuss how this notion of mirror symmetry may be thought of as a degeneration of mirror symmetry for lattice polarized K3's.