Virtual invariants from the non-associative Hilbert scheme

TL;DR

This paper introduces a non-associative model for the Hilbert scheme of points, providing a new framework with canonical virtual classes and explicit formulas, extending previous models to arbitrary dimensions.

Contribution

It constructs a non-associative Hilbert scheme with canonical virtual classes, offering explicit formulas and extending the scope of virtual class constructions to all dimensions.

Findings

Defined a smooth ambient space containing the classical Hilbert scheme.

Derived explicit formulas for virtual classes using localization and residues.

Extended virtual class constructions to all dimensions, including large n cases.

Abstract

We introduce a non-associative model for the Hilbert scheme of points in arbitrary dimension. We define a smooth ambient space, which we call the non-associative Hilbert scheme, containing the classical nested Hilbert scheme $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ as the associativity, cut out by an explicit section of an associativity bundle. This construction yields canonical perfect obstruction theories and virtual fundamental classes on $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ for all $(n,\underline d)$. Using virtual localization, we obtain closed formulas for these virtual classes as sums over admissible nested partitions. Over the punctual locus, we rewrite these as a single multivariable iterated residue formula governing all virtual integrals. Our construction works for all $n$, produces positive-dimensional virtual classes when $n$ is large compared to the number of points, and we expect that they extend the non-commutative matrix model and virtual class construction on Calabi-Yau threefolds.