All Toric Hermitian ALE Gravitational Instantons

We prove that the only smooth, Ricci flat, ALE gravitational instanton with a toric Hermitian non-Kähler structure is the Eguchi-Hanson instanton. The proof is analogous to the classification of toric Hermitian ALF instantons by Biquard and Gauduchon, although we avoid the use of toric Kähler geometry and instead perform a direct global analysis of the Tod form of the metric in Weyl-Papapetrou coordinates. This supports a conjecture by Gibbons and Bando-Kasue-Nakajima which states that any Ricci flat ALE instanton is self-dual.