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 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 quadratically to flat $T^2$-submersions over flat 2-dimensional cones which 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 ${\mathrm{I}}_b$ and ${{\mathrm{I}}_b}^{\ast }$, 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 ${\mathrm{I}}_b$ examples show that the curvature estimate from Cheeger-Tian \cite{ct-einstein} cannot be improved in general.

57 pages; v2: decay theory in Section 3.4 now covers the I_b and I_b^* cases as well, various minor corrections; v3: corrected remarks on compactification of ALH spaces (Definition 1.9, Problem 6.1), rewrote Section 2.2, included references to recent work of Biquard and Minerbe (Remark 1.6(iv)), fixed a gap in Section 5.3 (see the new Step 3 there), additional minor corrections

References in corpus (10)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• Gravitational Instantons of Type D_k
• A Construction of Complete Ricci-flat Kähler Manifolds
• Deformations of asymptotically cylindrical G_2 manifolds
• Regularity of asymptotically conical Ricci-flat Kähler metrics
• Geometric Structures of Collapsing Riemannian Manifolds I
• Ricci-flat deformations of asymptotically cylindrical Calabi--Yau manifolds
• Calabi-Yau metrics on Kummer surfaces as a model glueing problem
• On some asymptotically flat manifolds with non maximal volume growth
• Geometric Structures of Collapsing Riemannian Manifolds II: N*-bundles and Almost Ricci Flat Spaces

Cited by in corpus (3)

• Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves
• Asymptotically conical Calabi-Yau metrics on quasi-projective varieties
• Steady gradient Kähler-Ricci solitons on crepant resolutions of Calabi-Yau cones