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 \ D admits complete Ricciflat Kähler metrics in most de Rham cohomology classes. If D is smooth, the metrics converge to split flat cylinders 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 crosssection 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 I b and 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 I b examples show that the curvature estimate from Cheeger-Tian [14] cannot be improved in general.

A Kummer Construction for Gravitational Instantons

“…Here n is the normal vector, T and S are linear operations which we do not need to write down explicitly, since from the control (12) and the fact that the volume of ∂D ρ is O(ρ 2 ), we obtain that all boundary integrals go to zero when ρ goes to infinity. Finally this implies the following form of the Hitchin-Thorpe inequality:…”

“…One can either solve directly on Xj or find a Z j -invariant solution on X j : this amounts to solving the Monge-Ampère equation for cylindrical ends, and we refer to [26,14,15]. More specifically the case of X j is done in [12].…”

“…These manifolds are of special interest in quantum gravity and string theory, hence some motivation to understand examples. Previous constructions of gravitational instantons were either explicit [8,9], based on hyperkähler reduction [16,7], gauge theory [2,5,6], twistorial methods [13,4] or on the Monge-Ampère method of Tian and Yau [26,27], see also [14,15,12,25].…”

“…For X 1 = R × T 3 /±, our construction yields an ALH gravitational instanton. These ALG and ALH examples have been constructed recently in a general Tian-Yau framework on rational elliptic surfaces in [12] (see also [15,25]); in [12], they correspond to isotrivial elliptic fibrations. Two other quotients,…”

Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

“…Using one of the Tian-Yau theorems [54, Proposition 4.1] (see [28,Proposition 3.1] for the precise statement that we need and for an exposition of the proof), we can therefore assert that (4.23) has a solution u ∈ C ∞ (M ) such that sup M |∇ k u| < ∞ with respect to ĝc for all k ∈ N 0 .…”