On the QALE geometry of Nakajima's metric

TL;DR

This paper proves that Nakajima's hyperk"ahler metric on the Hilbert scheme of points on a2^2a2 is equivalent to Joyce's QALE metric, establishing a geometric correspondence between two constructions.

Contribution

It demonstrates the equivalence of Nakajima's hyperk"ahler metric and Joyce's QALE metric on the Hilbert scheme of points, linking two different geometric approaches.

Findings

Nakajima's metric is QALE as constructed by Joyce

The hyperk"ahler metric matches Joyce's QALE metric on the Hilbert scheme

Establishes a geometric correspondence between two metric constructions

Abstract

We show that on Hilbert scheme of $n$ points on $\C^2$, the hyperk\"ahler metric construsted by H. Nakajima via hyperk\"ahler reduction is the Quasi-Asymptotically Locally Euclidean (QALE in short) metric constructed by D. Joyce.