  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.

Differential Geometry;Algebraic Geometry