Gravitational instantons and del Pezzo surfaces

We show that any ALH$^*$ gravitational instanton arises from the Tian-Yau construction on a weak del Pezzo surface minus an anticanonical elliptic curve. A corollary is that any ALH$^*$ gravitational instanton can be compactified to a rational elliptic surface with respect to a hyperkaehler rotation of the weak del Pezzo complex structure.