Virtual invariants from the non-associative Hilbert scheme

Technical Summary: Virtual Invariants from the Non-Associative Hilbert Scheme

Problem Statement
The geometry of Hilbert schemes of points, particularly in dimensions $n \geq 3$, is notoriously subtle. While the Hilbert scheme of points on a smooth surface is smooth and well-understood, the Hilbert scheme of points on a threefold or higher-dimensional variety is typically singular. In enumerative geometry, particularly for Calabi-Yau threefolds and fourfolds, one relies on virtual fundamental classes to define invariants (such as Donaldson-Thomas invariants). These classes are traditionally constructed via perfect obstruction theories, often realized as critical loci of functions on smooth moduli spaces of non-commuting matrices (the "non-commutative matrix model").

However, existing models face limitations:

1. Non-commutative models replace the commutative coordinate ring with a free associative algebra, realizing the Hilbert scheme as a critical locus of a potential enforcing commutativity relations.
2. Nested Hilbert schemes (chains of subschemes) are generally singular with poorly understood component structures, yet they are essential for sheaf counting theories (e.g., Vafa-Witten theory, Seiberg-Witten theory).
3. There is a need for a unified, elementary construction of virtual fundamental classes for nested Hilbert schemes in arbitrary dimensions that yields positive-dimensional virtual classes when the ambient dimension $n$ is large relative to the number of points $d$.

Methodology
The authors introduce a non-associative model for the Hilbert scheme of points. Instead of relaxing commutativity (as in non-commutative models), they relax associativity while maintaining commutativity.

1. Construction of the Non-Associative Ambient Space:
   The authors define the non-associative nested Hilbert scheme, denoted $na\text{NHilb}^{\mathbf{d}}(A^n)$, as the moduli space of chains of ideals $I_{r+1} \subset \dots \subset I_1 \subset I_0 = \mathbb{C}\{x_1, \dots, x_n\}^c$, where $\mathbb{C}\{x_1, \dots, x_n\}^c$ is the free unital, commutative, but non-associative algebra on $n$ generators.

   • They prove that $na\text{NHilb}^{\mathbf{d}}(A^n)$ is a smooth variety.
   • The construction utilizes a quotient stack description $[M_{n,d}/P_d]$, where $M_{n,d}$ is a smooth locally closed subscheme of a vector space parameterizing algebra structures, and $P_d$ is a parabolic group acting freely.

2. Realization of the Classical Scheme:
   The classical nested Hilbert scheme $NHilb^{\mathbf{d}}(A^n)$ (where the algebras are associative) is embedded into the non-associative ambient space as the associativity locus.

   • This locus is cut out by the vanishing of a global section $s_{ass}$ of a vector bundle $E_{ass}$ (the associativity bundle) over the smooth ambient space.
   • Consequently, $NHilb^{\mathbf{d}}(A^n)$ inherits a canonical perfect obstruction theory and a virtual fundamental class $[NHilb^{\mathbf{d}}(A^n)]^{vir}$.

3. Virtual Localization:
   The authors utilize the torus action $T = (\mathbb{C}^*)^n$ on $A^n$, which lifts to the non-associative Hilbert scheme. They apply the Graber-Pandharipande virtual localization formula to compute the virtual class.

   • The fixed points correspond to chains of monomial ideals, which are in bijection with nested partitions.
   • The authors derive explicit formulas for the tangent space and the obstruction bundle at these fixed points, expressed in terms of the weights of the torus action.

4. Iterated Residue Formulas:
   For the punctual locus (specifically the nilpotently filtered locus $NHilb^{\mathbf{d}}_{nil\text{-}fil}(A^n)$), the authors construct a partial resolution fibering over a flag variety. Using Atiyah-Bott localization on this resolution, they transform the virtual integrals into iterated residue formulas.

   • A key technical result is that for the nilpotently filtered locus, all residues vanish except for a single contribution from the "Porteous" nested partition (containing only unit vectors in natural order).

Key Contributions and Results

• Theorem 1.1 (Virtual Localization Formula): The authors provide an explicit formula for the equivariant virtual fundamental class $[NHilb^{\mathbf{d}}(A^n)]^{vir}$ as a sum over admissible nested partitions. The weights are given by rational functions of the torus weights $s_i$.

  • Admissibility Condition: A partition is admissible if it does not contain vectors with 3 or more non-zero coordinates, and its projection to any 2D plane lies within specific bounds (related to the "Porteous" condition).
  • This formula allows for the computation of virtual invariants for all pairs $(n, d)$.

• Theorem 1.2 (Iterated Residue Formula): For the nilpotently filtered locus with dimension vector $\mathbf{d} = (1, d_1, \dots, d_r)$, the authors derive a single multivariable iterated residue formula for virtual integrals of tautological classes.

  • The formula is valid even when the total length $d$ exceeds the ambient dimension $n$, achieved by embedding the problem into a higher-dimensional space $A^N$ ($N \gg n$) and comparing virtual structures.
  • The formula generalizes earlier techniques by Bérczi and Szenes used in singularity theory and Thom polynomial computations.

• Positive-Dimensional Virtual Classes: Unlike many previous constructions where virtual classes are zero-dimensional (e.g., on Calabi-Yau threefolds), this construction produces positive-dimensional virtual classes whenever the ambient dimension $n$ is sufficiently large compared to the number of points.

• Relation to Non-Commutative Models: The paper establishes a rational map from the non-associative Hilbert scheme to the non-commutative nested Hilbert scheme. This map is well-defined on the domain containing the nested Hilbert scheme of points and restricts to the identity on the associativity locus. This suggests a bridge between the non-associative (commutative but non-associative) and non-commutative approaches to DT invariants.

Significance and Claims
The authors claim that this work provides an elementary and concrete construction of virtual fundamental classes for nested Hilbert schemes in all dimensions.

• Unification: It unifies the study of tautological intersection numbers (previously studied on the curvilinear component) with virtual intersection theory on the full Hilbert scheme.
• Combinatorial Control: The theory demonstrates that universal intersection numbers on Hilbert schemes of points are governed by explicit multi-variable residue formulas over the combinatorial data of monomial ideals and partitions, mirroring the structure of tautological theories.
• Extension to Higher Dimensions: The construction is expected to extend the non-commutative matrix model and virtual class constructions on Calabi-Yau threefolds to higher dimensions and nested settings.
• Future Work: The authors note that in forthcoming work, this framework will be used to reconstruct Donaldson-Thomas invariants of Calabi-Yau threefolds from the non-associative model, revealing a direct bridge between tautological and virtual theories.

The paper does not claim to solve the general existence of DT invariants for all sheaf moduli spaces but rather provides a specific, robust machinery for computing virtual invariants on nested Hilbert schemes of points using a novel non-associative geometric model.