[Paper Review] Complete Calabi-Yau metrics from P^2 # 9 \bar P^2

Hans‐Joachim Hein|arXiv (Cornell University)|Mar 12, 2010

Geometry and complex manifolds64 references6 citations

TL;DR

This paper constructs complete Ricci-flat Kähler metrics on the complement of a fiber in the rational elliptic surface obtained by blowing up ℙ² at nine base points of a pencil of cubics. Using a Tian-Yau PDE method with a Gross-Wilson-inspired ansatz, it proves existence in most de Rham cohomology classes, with precise asymptotic behavior depending on the monodromy of the fiber: exponential decay to a split cylinder for smooth fibers, quadratic decay to a flat T²-submersion for finite monodromy, and power-law decay for infinite monodromy, including sharp curvature estimates that show the Cheeger-Tian bound cannot be improved.

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 $\R^+ imes S^1 imes D$ 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 ${ m I}_b$ and ${{ m I}_b}^*$, 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 ${ m I}_b$ examples show that the curvature estimate from Cheeger-Tian \cite{ct-einstein} cannot be improved in general.

Motivation & Objective

• To construct complete Ricci-flat Kähler metrics on the complement of a fiber in the rational elliptic surface X = ℙ²#9ℙ̄².
• To understand the asymptotic geometry of these metrics based on the monodromy type of the fiber D.
• To establish sharp curvature decay estimates and prove a partial uniqueness result in the cylindrical case.
• To describe the local structure of the Einstein moduli space near cylindrical metrics with non-split cross-sections.
• To investigate whether the curvature decay estimates from Cheeger-Tian can be improved, particularly in the presence of singular fibers.

Proposed method

• Uses a Tian-Yau PDE method to solve a complex Monge-Ampère equation on X\D with a weighted Sobolev inequality framework.
• Employs an ansatz for the metric inspired by Gross-Wilson's work on singular fibers in elliptic fibrations.
• Applies weighted Sobolev inequalities and isoperimetric estimates to control the behavior of solutions near infinity and the fiber.
• Solves an ε-perturbed complex Monge-Ampère equation and takes the limit as ε→0 to obtain a complete Ricci-flat metric.
• Analyzes decay rates of curvature and volume growth by classifying the fiber type (smooth, finite monodromy, infinite monodromy).
• Uses the implicit function theorem and linearized analysis to study moduli space structure and prove uniqueness in the cylindrical case.

Experimental results

Research questions

• RQ1 Do complete Ricci-flat Kähler metrics exist on X\D for most de Rham cohomology classes, where X is the blowup of ℙ² at nine base points of a pencil of cubics and D is a fiber of the resulting elliptic fibration?
• RQ2 What is the precise asymptotic geometry of these metrics when D is smooth, and how fast do they converge to a split flat cylinder?
• RQ3 Can the curvature decay estimates from Cheeger-Tian be improved, and what do they reveal about the geometry of singular fibers with infinite monodromy?
• RQ4 What is the local structure of the Einstein moduli space near cylindrical metrics whose cross-section does not split off a circle?
• RQ5 Do the metrics exhibit different volume growth and curvature decay rates for Kodaira types Ib and I*b, and if so, what are the exact rates?

Key findings

• For smooth fibers D, the complete Ricci-flat metrics on X\D converge to a split flat cylinder ℝ⁺×S¹×D at an exponential rate.
• For fibers of finite monodromy, the metrics converge quadratically to flat T²-submersions over flat 2D cones, which may not be quotients of ℝ².
• For Kodaira type Ib fibers, the volume growth is r⁴/³ and curvature decays like r⁻²; for I*b fibers, volume growth is r² and curvature decays like r⁻²(log r)⁻¹.
• The curvature decay rates for Ib fibers show that the Cheeger-Tian estimate cannot be improved in general, as the decay is sharp.
• In the cylindrical case with finite monodromy, a partial uniqueness result holds, and the Einstein moduli space admits a local description including non-split cross-sections.
• The paper constructs a linear map between deformation spaces and proves it is an isomorphism, confirming the local structure of the moduli space near cylindrical metrics.