Recent progress on SYZ mirror symmetry for some non-compact Calabi-Yau surfaces

TL;DR

This paper surveys recent advances in SYZ mirror symmetry for certain non-compact Calabi-Yau surfaces, highlighting applications to gravitational instantons and their compactifications.

Contribution

It presents new results on SYZ mirror symmetry for rational elliptic and del Pezzo surfaces, including applications to gravitational instantons.

Findings

Proved Torelli theorem for $ALH^*$ gravitational instantons.

Showed all $ALH^*$ instantons can be compactified to weak del Pezzo surfaces.

Connected mirror symmetry results to gravitational instanton classification.

Abstract

We survey the authors recent works, joint with A. Jacob, on Strominger-Yau-Zaslow mirror symmetry for rational elliptic surfaces and del Pezzo surfaces. We discuss some applications, including the Torelli theorem for $ALH^*$ gravitational instantons and explain how our results can be used to prove that all $ALH^*$ gravitational instantons can be compactified to a weak del Pezzo surface, recovering a recent result of Hein-Sun-Viaclovsky-Zhang.