Gravitational instantons and special geometry

Abstract

The Chen–Teo gravitational instanton is an asymptotically flat, toric, Ricci-flat family of metrics on $\mathbb{CP}^2 \setminus S^1$, that provides a counter-example to the classical Euclidean Black Hole Uniqueness conjecture. In this paper we show that the Chen–Teo instanton is Hermitian and non-Kähler. It follows that all known examples of gravitational instantons are Hermitian.