Mirror Symmetry for Lattice Polarized del Pezzo Surfaces

TL;DR

This paper introduces a lattice polarization framework for rational elliptic and weak del Pezzo surfaces, proposing a mirror symmetry correspondence akin to Fano-LG duality and connecting it to existing mirror symmetry theories.

Contribution

It develops a new notion of lattice polarization for these surfaces and formulates a mirror symmetry relation, extending existing theories like Dolgachev-Nikulin-Pinkham and Gross-Siebert.

Findings

Describes the complex moduli of rational elliptic surfaces.

Characterizes the Kähler cone of weak del Pezzo surfaces.

Proposes a mirror symmetry relating these two geometric objects.

Abstract

We describe a notion of lattice polarization for rational elliptic surfaces and weak del Pezzo surfaces, and describe the complex moduli of the former and the K\"{a}hler cone of the latter. We then propose a version of mirror symmetry relating these two objects, which should be thought of as a form of Fano-LG correspondence. Finally, we relate this notion to other forms of mirror symmetry, including Dolgachev-Nikulin-Pinkham mirror symmetry for lattice polarized K3 surfaces and the Gross-Siebert program.