EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat Kähler ALE metrics for several infinite families of Kähler ALE spaces.

Asymptotically conical Calabi-Yau metrics with cone singularities along a compact divisor

We construct Asymptotically Locally Euclidean (ALE) and, more generally, asymptotically conical Calabi–Yau metrics with cone singularities along a compact simple normal crossing divisor. In particular, this includes the case of the minimal resolution of 2D quotient singularities for any finite subgroup $\Gamma \subset U(2)$ acting freely on the three-sphere, hence generalizing Kronheimer’s construction of smooth ALE gravitational instantons.

On the quasi-asymptotically locally Euclidean geometry of Nakajima's metric

We show that on the Hilbert scheme of n points on ℂ2, the hyperkähler metric constructed by Nakajima via hyperkähler reduction is the quasi-asymptotically locally Euclidean (QALE) metric constructed by Joyce.

SUPERSTRINGS AND QUANTUM GRAVITY

It is suggested that the quantum geometry of asymptotically locally Euclidean spaces underlies the dynamics of superstrings moving on Calabi-Yau manifolds. In particular, it is pointed out that solving the Wheeler-DeWitt equation is akin to using a neural network to color maps on a Riemann surface. In the infrared limit such map colorings are formally equivalent to Calabi-Yau vacua for superstrings.