Gravitational instantons from rational elliptic surfaces

Abstract:

Let $X$ denote the complex projective plane, blown up at the nine base points of a pencil of cubics, and let $D$ be any fiber of the resulting elliptic fibration on $X$. Using ansatz metrics inspired by work of Gross-Wilson and a PDE method due to Tian-Yau, we prove that $X \setminus D$ admits complete Ricci-flat Kähler metrics in most de Rham cohomology classes. If $D$ is smooth, the metrics converge to split flat cylinders $\mathbb {R}^+ \times S^1 \times D$ 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 $D$ is singular but of finite monodromy, they converge at least polynomially to flat $T^2$-submersions over flat $2$-dimensional cones that need not be quotients of $\mathbb {R}^2$. If $D$ is singular of infinite monodromy, their volume growth rates are $4/3$ and $2$ for the Kodaira types $\textrm {I}_b$ and ${\textrm {I}_b}^*$, their injectivity radii decay like $r^{-1/3}$ and $(\log r)^{-1/2}$, and their curvature tensors decay like $r^{-2}$ and $r^{-2}(\log r)^{-1}$. In particular, the $\textrm {I}_b$ examples show that a curvature estimate due to Cheeger and Tian cannot be improved in general.