Complete Calabi-Yau metrics from P^2 # 9 \bar P^2

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 K\"ahler metrics in most de Rham cohomology classes. If $D$ is smooth, the metrics converge to split flat cylinders $\R^+ \times S^1 \times D$ 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 quadratically to flat $T^2$-submersions over flat 2-dimensional cones which need not be quotients of $\R^2$. If $D$ is singular of infinite monodromy, their volume growth rates are 4/3 and 2 for the Kodaira types ${\rm I}_b$ and ${{\rm I}_b}^*$, 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 ${\rm I}_b$ examples show that the curvature estimate from Cheeger-Tian \cite{ct-einstein} cannot be improved in general.