Abstract

Gravitational instantons are the building blocks of Hawking's quantum gravity theory. They are non-compact hyperK\"ahler 4-manifolds with L^2 curvature and can be viewed as the local pieces of K3 surfaces. Similar to K3 surfaces, the cohomology classes of the hyperK\"ahler triples of gravitational instantons induce a period map. In this talk, we will restrict to a particular type of gravitational instanton of type ALH^*. As a side product of studying SYZ mirror symmetry of log Calabi-Yau surfaces, we will prove that the period map is both injective and surjective. As a consequence, the period domain is the moduli space of ALH^*-gravitational instantons. The talk is based on joint works with T. Collins, A. Jacob and the joint work with T.-J. Lee.