The Torelli Theorem for $ALH^*$ Gravitational Instantons

TL;DR

This paper provides a concise proof of the Torelli theorem for $ALH^*$ gravitational instantons, leveraging mirror symmetry constructions and recent advances to classify their diffeomorphism types.

Contribution

It offers a new, streamlined proof of the Torelli theorem for $ALH^*$ instantons and identifies ten distinct diffeomorphism types.

Findings

Proof of Torelli theorem for $ALH^*$ instantons

Classification of 10 diffeomorphism types

Application of mirror symmetry techniques

Abstract

We give a short proof of the Torelli theorem for $ALH^*$ gravitational instantons, using the authors' previous construction of mirror special Lagrangian fibrations in del Pezzo surfaces and rational elliptic surfaces together with recent work of Sun-Zhang. In particular, this includes an identification of 10 diffeomorphism types of $ALH^*_b$ gravitational instantons.