Torelli-type theorems for gravitational instantons with quadratic volume growth

TL;DR

This paper establishes Torelli-type theorems for certain gravitational instantons, showing that their periods uniquely determine them up to diffeomorphism, and explores degenerations of hyperk"ahler metrics on K3 surfaces.

Contribution

It proves Torelli-type uniqueness theorems for ALG and ALG$^*$ gravitational instantons, and constructs new degenerations of hyperk"ahler metrics on K3 surfaces.

Findings

Periods uniquely characterize ALG and ALG$^*$ gravitational instantons.

The period map is surjective for ALG and open for ALG$^*$ cases.

New degenerations of hyperk"ahler metrics on K3 surfaces exhibiting bubbling of ALG$^*$ instantons.

Abstract

We prove Torelli-type uniqueness theorems for both ALG$^*$ gravitational instantons and ALG gravitational instantons which are of order $2$. That is, the periods uniquely characterize these types of gravitational instantons up to diffeomorphism. We define a period mapping $\mathscr{P}$, which we show is surjective in the ALG cases, and open in the ALG$^*$ cases. We also construct some new degenerations of hyperk\"ahler metrics on the K3 surface which exhibit bubbling of ALG$^*$ gravitational instantons.