Gravitational instantons from rational elliptic surfaces

Let X X denote the complex projective plane, blown up at the nine base points of a pencil of cubics, and let D D be any fiber of the resulting elliptic fibration on X X . Using ansatz metrics inspired by work of Gross-Wilson and a PDE method due to Tian-Yau, we prove that X ∖ D X D admits complete Ricci-flat Kähler metrics in most de Rham cohomology classes. If D D is smooth, the metrics converge to split flat cylinders R + × S 1 × D R^+ × S^1 × D at an exponential rate. In this case, we also obtain a partial uniqueness result and a local description of the Einstein moduli space, which contains cylindrical metrics whose cross section does not split off a circle. If D D is singular but of finite monodromy, they converge at least polynomially to flat T 2 T^2 -submersions over flat 2 2 -dimensional cones that need not be quotients of R 2 R^2 . If D D is singular of infinite monodromy, their volume growth rates are 4 / 3 4/3 and 2 2 for the Kodaira types I b I_b and