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arxiv: 2606.30172 · v1 · pith:U5YS2DTNnew · submitted 2026-06-29 · 🧮 math.AG

Virtual K-theoretic invariants of the nested Hilbert scheme on mathbb{C}²

Felix Minddal This is my paper

Pith reviewed 2026-06-30 03:40 UTC · model grok-4.3

classification 🧮 math.AG
keywords nested Hilbert schemevirtual structure sheafK-theoretic invariantsnon-commutative Hilbert schemeequivariant localizationperfect obstruction theorygenerating seriesHilbert scheme of points
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The pith

The pushforward of the virtual structure sheaf on the nested Hilbert scheme is a constant twist of the one on the lower level.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a nested non-commutative Hilbert scheme on affine space and realizes the ordinary nested Hilbert scheme of points as the locus where two families of matrices commute. In the two-dimensional case this locus is exhibited as the common zero set of two global sections of vector bundles on the non-commutative space, which supplies a perfect obstruction theory. The central technical step is to produce a map of these bundles that is compatible with the forgetful morphism to the lower nesting level; the induced map on virtual structure sheaves is then shown to be multiplication by a single equivariant class. Localization reduces this class to the equivariant Euler characteristic of a tautological bundle on the ordinary Hilbert scheme of points, yielding an explicit closed formula for the full multivariate generating series of the virtual invariants.

Core claim

Using a map of bundles on the non-commutative Hilbert scheme, the torus-equivariant pushforward of the virtual structure sheaf under the forgetting map is a twist of the virtual structure sheaf on the lower level by a constant class given by the equivariant Euler characteristic of a tautological class of the Hilbert scheme of points, from which a closed formula for the multivariate generating series of the equivariant virtual Euler characteristic of the nested Hilbert scheme follows.

What carries the argument

The nested non-commutative Hilbert scheme, whose commutativity locus is realized as the zero locus of two sections of bundles and thereby carries a perfect obstruction theory equivalent to the one of Gholampour–Sheshmani–Yau.

If this is right

  • The virtual K-theoretic invariants of the nested scheme satisfy a recursive relation given by multiplication with the constant twist class.
  • The multivariate generating series of these invariants admits an explicit closed-form expression.
  • The obstruction theory constructed via the non-commutative model coincides with the one previously obtained by Gholampour, Sheshmani and Yau.
  • Equivariant localization reduces all computations to fixed-point contributions on the ordinary Hilbert scheme of points.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bundle-map technique may supply obstruction theories for nested schemes in dimensions greater than two where direct constructions are more difficult.
  • The appearance of a universal constant twist suggests that higher-nesting invariants are completely determined by ordinary Hilbert-scheme invariants once the tautological Euler class is known.
  • Analogous maps of bundles could be used to relate virtual K-theoretic invariants across other moduli spaces that admit non-commutative presentations.

Load-bearing premise

The commutativity locus on the non-commutative Hilbert scheme can be described as the zero locus of two specific bundle sections, and this description produces a perfect obstruction theory matching the known one.

What would settle it

A direct localization computation of the virtual Euler characteristic for the nested pair of lengths (1,2) that fails to equal the product of the lower-level invariant and the predicted constant twist factor coming from the tautological Euler class on the length-1 Hilbert scheme.

read the original abstract

We construct a nested version of the non-commutative Hilbert scheme and embed the nested Hilbert scheme of points on $\mathbb{C}^n$ as the commutativity locus. In the $\mathbb{C}^2$-case, we exhibit this locus as the zero locus of two different sections of bundles and use this description to equip the nested Hilbert scheme of points with a perfect obstruction theory equivalent to that of Gholampour, Sheshmani and Yau. We study the torus equivariant pushforward of the virtual structure sheaf under the map of nested Hilbert schemes forgetting the largest subscheme of the nesting. Using a map of the bundles on the non-commutative Hilbert scheme, we prove that this pushforward is a twist of the virtual structure sheaf on the lower level. Using localization, we show that the twist is by a constant class with values corresponding to the equivariant Euler characteristic of a tautological class of the Hilbert scheme of points. From this, we derive a closed formula for the multivariate generating series of the equivariant virtual Euler characteristic of the nested Hilbert scheme of points.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs a nested non-commutative Hilbert scheme and realizes the nested Hilbert scheme of points on ℝ^n as its commutativity locus. In the ℝ^2 case, this locus is exhibited as the zero locus of two sections of bundles, yielding a perfect obstruction theory equivalent to that of Gholampour-Sheshmani-Yau. The torus-equivariant pushforward of the virtual structure sheaf under the forgetting map (to the lower nested level) is shown to be a twist of the lower virtual structure sheaf via a bundle map; localization identifies the twist factor as a constant class given by the equivariant Euler characteristic of a tautological bundle on the ordinary Hilbert scheme of points. This produces a closed formula for the multivariate generating series of the equivariant virtual Euler characteristics of the nested Hilbert schemes.

Significance. If the obstruction-theory equivalence and the subsequent pushforward and localization steps hold, the closed formula for the generating series constitutes a concrete computational advance in virtual K-theory of moduli spaces of nested subschemes. The non-commutative Hilbert scheme construction supplies an explicit geometric model that may extend to other nested moduli problems.

major comments (2)
  1. [Abstract and obstruction-theory section] Abstract (C^2 paragraph) and the section establishing the obstruction theory: the claim that the commutativity locus, realized as the zero locus of two sections of bundles on the non-commutative Hilbert scheme, carries a perfect obstruction theory equivalent to Gholampour-Sheshmani-Yau is asserted without an explicit comparison of the obstruction sheaves or verification that the virtual structure sheaves coincide in K-theory; this equivalence is load-bearing for the identification of the virtual structure sheaf used in the pushforward argument, the twist proof, and the localization computation that yields the closed formula.
  2. [Pushforward and localization section] Section on the pushforward and localization: the statement that the twist factor is a constant class whose value is the equivariant Euler characteristic of a tautological class on the Hilbert scheme of points requires an explicit check that no higher cohomology or non-constant terms survive after localization; without this, the reduction to a closed multivariate generating series is not fully justified.
minor comments (2)
  1. [Construction section] Clarify the precise definition of the nested non-commutative Hilbert scheme and the two sections whose common zero locus is used; the notation for the bundles and the forgetting map should be introduced with explicit diagrams or equations.
  2. [Formula derivation] Add a short table or explicit low-degree examples comparing the new formula with known values of virtual Euler characteristics for small n to make the closed formula more immediately verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. The comments highlight areas where the arguments can be made more explicit, and we address each major comment below. We will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and obstruction-theory section] Abstract (C^2 paragraph) and the section establishing the obstruction theory: the claim that the commutativity locus, realized as the zero locus of two sections of bundles on the non-commutative Hilbert scheme, carries a perfect obstruction theory equivalent to Gholampour-Sheshmani-Yau is asserted without an explicit comparison of the obstruction sheaves or verification that the virtual structure sheaves coincide in K-theory; this equivalence is load-bearing for the identification of the virtual structure sheaf used in the pushforward argument, the twist proof, and the localization computation that yields the closed formula.

    Authors: We agree that an explicit comparison of the obstruction sheaves and confirmation that the virtual structure sheaves coincide in K-theory would make the equivalence fully rigorous. The current argument relies on the zero-locus description and standard properties of obstruction theories for such loci, but we acknowledge the need for a direct verification. In the revised manuscript, we will add a dedicated subsection providing this comparison, including the relevant sheaf diagrams and K-theory identification. This will support the subsequent pushforward and localization steps. revision: yes

  2. Referee: [Pushforward and localization section] Section on the pushforward and localization: the statement that the twist factor is a constant class whose value is the equivariant Euler characteristic of a tautological class on the Hilbert scheme of points requires an explicit check that no higher cohomology or non-constant terms survive after localization; without this, the reduction to a closed multivariate generating series is not fully justified.

    Authors: The localization in the paper proceeds via torus-fixed loci on the nested Hilbert scheme, where the relevant tautological bundles yield contributions that reduce to constants by the structure of the equivariant cohomology ring and vanishing results on the fixed components. However, to address the concern directly, we will expand the localization subsection to include an explicit verification that higher cohomology groups vanish and that no non-constant terms remain after localization. This will justify the reduction to the closed generating series formula. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external GSY obstruction theory and standard localization

full rationale

The paper constructs the nested non-commutative Hilbert scheme and realizes the nested Hilbert scheme of points as a commutativity locus realized as zero locus of two bundle sections. It then equips this locus with a perfect obstruction theory shown equivalent to the external reference Gholampour-Sheshmani-Yau via that explicit description. The pushforward of the virtual structure sheaf, the identification of the twist factor via localization as the equivariant Euler characteristic of a tautological class on the ordinary Hilbert scheme, and the resulting closed formula for the multivariate generating series are derived from these constructions plus torus localization; none reduce by definition or self-citation to the paper's own fitted inputs. The cited GSY result is independent and external. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit list of free parameters, axioms, or invented entities; the construction implicitly relies on standard algebraic geometry background (torus actions, virtual classes) whose details are not visible.

pith-pipeline@v0.9.1-grok · 5712 in / 1327 out tokens · 38463 ms · 2026-06-30T03:40:12.202758+00:00 · methodology

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Reference graph

Works this paper leans on

46 extracted references · 13 canonical work pages

  1. [1]

    Knot homology and sheaves on the H ilbert scheme of points on the plane

    Oblomkov, Alexei and Rozansky, Lev. Knot homology and sheaves on the H ilbert scheme of points on the plane. Sel. Math. New Ser

  2. [2]

    M. V. Nori , title =. Proceedings of the International Symposium on Algebraic Geometry (Kyoto, 1977) , pages =. 1978 , publisher =

    1977

  3. [3]

    arxiv preprint , volume =

    Maxim Kazarian , title =. arxiv preprint , volume =. 2017 , url =

    2017

  4. [4]

    Gustavsen and D

    T.S. Gustavsen and D. Laksov and R.M. Skjelnes , abstract =. An elementary, explicit, proof of the existence of. Journal of Pure and Applied Algebra , volume =. 2007 , issn =. doi:https://doi.org/10.1016/j.jpaa.2006.11.003 , url =

    work page doi:10.1016/j.jpaa.2006.11.003 2007

  5. [5]

    Eugene Gorsky and Andrei Neguţ and Jacob Rasmussen , keywords =. Flag. Advances in Mathematics , volume =. 2021 , issn =. doi:https://doi.org/10.1016/j.aim.2020.107542 , url =

    work page doi:10.1016/j.aim.2020.107542 2021

  6. [6]

    Amin Gholampour and Artan Sheshmani and Shing-Tung Yau , keywords =. Nested. Advances in Mathematics , volume =. 2020 , issn =. doi:https://doi.org/10.1016/j.aim.2020.107046 , url =

    work page doi:10.1016/j.aim.2020.107046 2020

  7. [7]

    Thomas , journal=

    Amin Gholampour and Richard P. Thomas , journal=. Degeneracy loci, virtual cycles and nested. 2017 , url=

    2017

  8. [8]

    1998 , isbn =

    William Fulton , title =. 1998 , isbn =

    1998

  9. [9]

    Cohomology of Noncommutative H ilbert Schemes

    Reineke, Markus. Cohomology of Noncommutative H ilbert Schemes. Algebr. Represent. Theory

  10. [10]

    Motivic degree zero

    Behrend, Kai and Bryan, Jim and Szendr. Motivic degree zero. Inventiones mathematicae , volume=. 2013 , publisher=

    2013

  11. [11]

    Flags of sheaves, quivers and symmetric polynomials , volume=

    Bonelli, Giulio and Fasola, Nadir and Tanzini, Alessandro , year=. Flags of sheaves, quivers and symmetric polynomials , volume=. doi:10.1017/fms.2024.43 , journal=

    work page doi:10.1017/fms.2024.43 2024

  12. [12]

    T autological sheaves on H ilbert schemes of points

    Wang, Zhilan and Zhou, Jian. T autological sheaves on H ilbert schemes of points. J. Algebraic Geom

  13. [13]

    Wachs , keywords =

    John Shareshian and Michelle L. Wachs , keywords =. Chromatic quasisymmetric functions , journal =. 2016 , issn =. doi:https://doi.org/10.1016/j.aim.2015.12.018 , url =

    work page doi:10.1016/j.aim.2015.12.018 2016

  14. [14]

    Stanley , abstract =

    R.P. Stanley , abstract =. A Symmetric Function Generalization of the Chromatic Polynomial of a Graph , journal =. 1995 , issn =. doi:https://doi.org/10.1006/aima.1995.1020 , url =

    work page doi:10.1006/aima.1995.1020 1995

  15. [15]

    2025 , eprint=

    A proof of the Stanley--Stembridge conjecture , author=. 2025 , eprint=

    2025

  16. [16]

    Inventiones Mathematicae , volume =

    Behrend, Kai and Fantechi, Barbara , title =. Inventiones Mathematicae , volume =. 1997 , doi =

    1997

  17. [17]

    1971 , address =

    Luc Illusie , title =. 1971 , address =

    1971

  18. [18]

    Riemann--Roch theorems and elliptic genus for virtually smooth schemes

    Fantechi, Barbara and G \"o ttsche, Lothar. Riemann--Roch theorems and elliptic genus for virtually smooth schemes. Geom. Topol

  19. [19]

    Virtual fundamental classes, global normal cones and Fulton's canonical classes

    Siebert, Bernd. Virtual fundamental classes, global normal cones and Fulton's canonical classes. Frobenius Manifolds: Quantum Cohomology and Singularities. 2004. doi:10.1007/978-3-322-80236-1_13

    work page doi:10.1007/978-3-322-80236-1_13 2004

  20. [20]

    A K -theoretic F ulton class

    Thomas, Richard P. A K -theoretic F ulton class. Facets of Algebraic Geometry

  21. [21]

    Geometry of Moduli Spaces and Representation Theory , series =

    Andrei Okounkov , title =. Geometry of Moduli Spaces and Representation Theory , series =. 2017 , publisher =

    2017

  22. [22]

    Inventiones Mathematicae , volume =

    Graber, Tom and Pandharipande, Rahul , title =. Inventiones Mathematicae , volume =

  23. [23]

    Une formule de L efschetz en K -th \'e orie \'e quivariante alg \'e brique

    Thomason, R W. Une formule de L efschetz en K -th \'e orie \'e quivariante alg \'e brique. Duke Math. J

  24. [24]

    Virtual pullbacks in K -theory

    Qu, Feng. Virtual pullbacks in K -theory. Ann. Inst. Fourier (Grenoble)

  25. [25]

    Virtual pull-backs , volume =

    Manolache, Cristina , year =. Virtual pull-backs , volume =. Journal of Algebraic Geometry , publisher =. doi:10.1090/s1056-3911-2011-00606-1 , number =

    work page doi:10.1090/s1056-3911-2011-00606-1 2011

  26. [26]

    and Ricolfi, Andrea T

    Beentjes, Sjoerd V. and Ricolfi, Andrea T. , year=. Virtual counts on. Mathematical Research Letters , publisher=. doi:10.4310/mrl.2021.v28.n4.a2 , number=

    work page doi:10.4310/mrl.2021.v28.n4.a2 2021

  27. [27]

    Fasola, Nadir and Monavari, Sergej and Ricolfi, Andrea T. , year=. Higher rank. doi:10.1017/fms.2021.4 , journal=

    work page doi:10.1017/fms.2021.4 2021

  28. [28]

    2025 , eprint=

    Proof of a magnificent conjecture , author=. 2025 , eprint=

    2025

  29. [29]

    and Kononov, Y

    Arbesfeld, N. and Kononov, Y. , title =

  30. [30]

    Journal of High Energy Physics , year=

    Quiver Matrix Model and Topological Partition Function in Six Dimensions , author=. Journal of High Energy Physics , year=

  31. [31]

    Thomas , journal=

    Jeongseok Oh and Richard P. Thomas , journal=. Counting sheaves on. 2020 , url=

    2020

  32. [32]

    Magnificent Four with Colors , volume=

    Nekrasov, Nikita and Piazzalunga, Nicolò , year=. Magnificent Four with Colors , volume=. Communications in Mathematical Physics , publisher=. doi:10.1007/s00220-019-03426-3 , number=

    work page doi:10.1007/s00220-019-03426-3

  33. [33]

    Exts and vertex operators , volume=

    Carlsson, Erik and Okounkov, Andrei , year=. Exts and vertex operators , volume=. Duke Mathematical Journal , publisher=. doi:10.1215/00127094-1593380 , number=

    work page doi:10.1215/00127094-1593380

  34. [34]

    arXiv: Combinatorics , year=

    A modular relation for the chromatic symmetric functions of (3+1)-free posets , author=. arXiv: Combinatorics , year=

  35. [35]

    Algebraic Families on an Algebraic Surface , urldate =

    John Fogarty , journal =. Algebraic Families on an Algebraic Surface , urldate =

  36. [36]

    Open problems in deformations of

    Jelisiejew, Joachim , year =. Open problems in deformations of. doi:10.1090/conm/805/16122 , journal =

    work page doi:10.1090/conm/805/16122

  37. [37]

    Reducibility of the families of 0-dimensional schemes on a variety

    Iarrobino, A. Reducibility of the families of 0-dimensional schemes on a variety. Invent. Math

  38. [38]

    Pacific Journal of Mathematics , year=

    CELLULAR DECOMPOSITIONS FOR NESTED HILBERT SCHEMES OF POINTS , author=. Pacific Journal of Mathematics , year=

  39. [39]

    Journal of Algebra , year=

    Irreducibility and Singularities of Some Nested Hilbert Schemes , author=. Journal of Algebra , year=

  40. [40]

    Geometry & Topology , year=

    Universal polynomials for tautological integrals on Hilbert schemes , author=. Geometry & Topology , year=

  41. [41]

    2023 , eprint=

    Tautological integrals on Hilbert scheme of points II: Geometric subsets , author=. 2023 , eprint=

    2023

  42. [42]

    2023 , eprint=

    Tautological integrals on Hilbert scheme of points I , author=. 2023 , eprint=

    2023

  43. [43]

    Algebra & Number Theory , year=

    Symmetric Obstruction Theories and Hilbert Schemes of Points on Threefolds , author=. Algebra & Number Theory , year=

  44. [44]

    Thomas, R. P. A Holomorphic Casson invariant for Calabi-Yau three folds, and bundles on K3 fibrations. J. Diff. Geom. 2000. arXiv:math/9806111

    Pith/arXiv arXiv 2000

  45. [45]

    Mathematische Annalen , year=

    Hyperquot schemes on curves: virtual class and motivic invariants , author=. Mathematische Annalen , year=

  46. [46]

    2026 , eprint=

    Derived hyperquot schemes , author=. 2026 , eprint=

    2026