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

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 $AL{H}^{\ast }$ gravitational instantons and explain how our results can be used to prove that all $AL{H}^{\ast }$ gravitational instantons can be compactified to a weak del Pezzo surface, recovering a recent result of Hein-Sun-Viaclovsky-Zhang.

survey article; 29 pages