ALH Gravitational Instantons and Rational Surfaces

We construct complete ALH hyperkähler metrics on complements of anticanonical elliptic curves with zero self-intersection in smooth rational surfaces. This answers the existence question of Filip–Tosatti, related to the picture proposed by Haskins–Hein–Nordström. Conversely, we prove that every hyperkähler ALH gravitational instanton admits, for each induced complex structure, a compactification by an anticanonical elliptic curve in a rational surface.