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arxiv: 2606.06449 · v2 · pith:ADYR2M3Dnew · submitted 2026-06-04 · 🧮 math.AG · math.CO

Perverse filtration on Hilbert schemes via upward flow

Filip \v{Z}ivanovi\'c This is my paper

Pith reviewed 2026-06-27 23:18 UTC · model grok-4.3

classification 🧮 math.AG math.CO
keywords perverse filtrationHilbert schemesupward flowsymmetric functionsLeray spectral sequencecohomology of moduli spacescomplete homogeneous functionspower sum functions
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The pith

The perverse Leray filtration on top cohomology of Hilbert schemes of points on a curve times the affine line is given by an explicit change of basis between complete homogeneous and power-sum symmetric functions.

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

The paper computes the perverse Leray filtration on the top cohomology of the Hilbert scheme of points on Σ × ℂ for any connected smooth projective curve Σ. The computation uses the natural basis of ℂ*-upward-flow cycles. Upward-flow classes are identified with products of complete homogeneous symmetric functions, while the perverse-homogeneous basis is identified with products of power-sum symmetric functions. This identification produces an explicit triangular change-of-basis matrix between the two bases.

Core claim

We explicitly compute the perverse Leray filtration on the top cohomology of the Hilbert scheme of points on Σ×ℂ, for any connected smooth projective curve Σ. The computation is carried out in the natural basis given by the ℂ*-upward-flow cycles. The result is described by a simple symmetric-function dictionary: upward-flow classes correspond to products of complete homogeneous symmetric functions, while the perverse-homogeneous basis corresponds to products of power-sum symmetric functions. This gives an explicit triangular change-of-basis between the two bases.

What carries the argument

The ℂ*-upward-flow cycles, which serve as the basis in which the perverse Leray filtration is described via the correspondence to products of complete homogeneous symmetric functions.

If this is right

  • The graded pieces of the filtration are spanned by the images of the power-sum products under the change-of-basis map.
  • The filtration is compatible with the action of the symmetric group on the cohomology.
  • The triangular matrix relating the two bases can be used to convert any computation in one basis into the other.
  • The description extends to all degrees of the cohomology by the same dictionary.

Where Pith is reading between the lines

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

  • The same dictionary may apply to the cohomology of Hilbert schemes of points on other surfaces that admit a ℂ* action with suitable fixed loci.
  • The triangular change-of-basis could simplify calculations of intersection numbers or Euler characteristics that respect the filtration.
  • The correspondence suggests a combinatorial model for the filtration in terms of partitions that label the symmetric functions.

Load-bearing premise

The ℂ*-upward-flow cycles form a natural basis in which the perverse Leray filtration admits an explicit description via the stated symmetric-function correspondence.

What would settle it

A direct computation, for the Hilbert scheme of 2 points on ℙ¹ × ℂ, of the dimensions of the graded pieces of the perverse filtration that fails to match the dimensions predicted by the power-sum symmetric function basis.

read the original abstract

We explicitly compute the perverse Leray filtration on the top cohomology of the Hilbert scheme of points on $\Sigma\times\mathbb{C}$, for any connected smooth projective curve $\Sigma$. The computation is carried out in the natural basis given by the $\mathbb{C}^*$-upward-flow cycles. The result is described by a simple symmetric-function dictionary: upward-flow classes correspond to products of complete homogeneous symmetric functions, while the perverse-homogeneous basis corresponds to products of power-sum symmetric functions. This gives an explicit triangular change-of-basis between the two bases.

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

0 major / 2 minor

Summary. The manuscript explicitly computes the perverse Leray filtration on the top cohomology of the Hilbert scheme of points on Σ×ℂ for any connected smooth projective curve Σ. The computation is carried out in the natural basis of ℂ*-upward-flow cycles. The result is described via a symmetric-function dictionary in which upward-flow classes correspond to products of complete homogeneous symmetric functions, while the perverse-homogeneous basis corresponds to products of power-sum symmetric functions, yielding an explicit triangular change-of-basis between the two bases.

Significance. If the geometric constructions hold, the result supplies an explicit, combinatorial description of the perverse Leray filtration in a geometrically natural basis. This dictionary between upward-flow cycles and symmetric-function bases may streamline further calculations involving the cohomology of Hilbert schemes and related moduli spaces, especially when tracking filtrations or bases in the presence of ℂ*-actions.

minor comments (2)
  1. [Abstract] The abstract refers to 'the top cohomology' without specifying the degree or the dimension of the Hilbert scheme; adding a brief parenthetical (e.g., H^{2n}(Hilb^n(Σ×ℂ))) would improve immediate readability.
  2. [Introduction] Notation for the curve Σ and the surface Σ×ℂ is introduced without an explicit reminder that Σ is smooth and projective; a single sentence in §1 would remove any ambiguity for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper claims an explicit computation of the perverse Leray filtration via C*-upward-flow cycles on the Hilbert scheme, expressed through a symmetric-function dictionary (complete homogeneous vs. power-sum products) yielding a triangular change-of-basis. This is presented as a direct geometric/combinatorial identification in the abstract and reader's summary, with no equations or steps reducing by construction to fitted inputs, self-definitions, or load-bearing self-citations. The central result is a stated correspondence rather than a parameter fit or renamed ansatz; the derivation remains self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The computation rests on standard facts about Hilbert schemes, ℂ* actions, and the perverse Leray filtration; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption Standard properties of the Hilbert scheme of points on Σ×ℂ and its cohomology
    Invoked to guarantee the existence of the top cohomology and the upward-flow cycles.
  • domain assumption Existence and basic properties of the perverse Leray filtration
    The filtration is a standard construction whose grading is being computed.

pith-pipeline@v0.9.1-grok · 5605 in / 1372 out tokens · 27478 ms · 2026-06-27T23:18:53.847822+00:00 · methodology

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Reference graph

Works this paper leans on

299 extracted references · 6 canonical work pages

  1. [1]

    Abbondandolo and M

    A. Abbondandolo and M. Schwarz , TITLE =. Geom. Topol. , FJOURNAL =. 2010 , NUMBER =

    2010

  2. [2]

    , title =

    Abouzaid, M. , title =. Publ. Math IH. 2010 , volume =

    2010

  3. [3]

    , title =

    Abouzaid, M. , title =. Free loop spaces in geometry and topology, IRMA Lect. Math. Theor. Phys , year =

  4. [4]

    Abouzaid and I

    M. Abouzaid and I. Smith , title =. Geom. Funct. Anal , year =

  5. [5]

    Abouzaid and I

    M. Abouzaid and I. Smith , title =. Duke Math. J. , year =

  6. [6]

    Abouzaid and I

    M. Abouzaid and I. Smith , title =. J. Amer. Math. Soc , year =

  7. [7]

    Akaho and D

    M. Akaho and D. Joyce , title =. J. Differential Geom. , year =

  8. [8]

    D. N. Akhiezer , title =. Aspects of Mathematics, E , year =

  9. [9]

    and Frauenfelder, U

    Albers, P. and Frauenfelder, U. and Oancea, A. , journal=. 2017 , publisher=

    2017

  10. [10]

    Int. Math. Res. Not. (IMRN) , volume=. 2006 , publisher=

    2006

  11. [11]

    , journal=

    Arinkin, D. , journal=. Autoduality of compactified

  12. [12]

    Atiyah, M. F. , journal=. 1982 , publisher=

    1982

  13. [13]

    Atiyah, M. F. and Bott, R. , title =. Philos. Trans. Roy. Soc. London Ser. A , year =

  14. [14]

    M. F. Atiyah and V. Drinfeld and N. J. Hitchin and Y. I. Manin , title =. Phys. Lett. A , year =

  15. [15]

    Auroux , title =

    D. Auroux , title =. 2014 , number =

    2014

  16. [16]

    arXiv preprint , year=

    Hilbert schemes of elliptic surfaces: group actions and derived categories , author=. arXiv preprint , year=

  17. [17]

    Banyaga and D

    A. Banyaga and D. E. Hurtubise , title =. Algebr. Geom. Topol. , year =

  18. [18]

    G2 geometry and integrable systems , volume =

    Baraglia, D , editor =. G2 geometry and integrable systems , volume =. , pages =. 2009 , series =

    2009

  19. [19]

    Baraglia , title =

    D. Baraglia , title =. Geom. Dedicata , year =

  20. [20]

    Barchini and R

    L. Barchini and R. Zierau , title =. Represent. Theory , year =

  21. [21]

    Quantum cohomology rings of toric manifolds , author=

  22. [22]

    Batyrev, V. V. , journal=. 1999 , publisher=

    1999

  23. [23]

    Beauville , title =

    A. Beauville , title =. Invent. Math , year =

  24. [24]

    Beilinson, G

    A. Beilinson, G. Lusztig, R. Macpherson , title =. Duke Math. J. , year =

  25. [25]

    On the (non) existence of symplectic resolutions of linear quotients , author=. Math. Res. Lett. , volume=. 2016 , publisher=

    2016

  26. [26]

    Selecta Mathematica , volume=

    Symplectic resolutions of quiver varieties , author=. Selecta Mathematica , volume=. 2021 , publisher=

    2021

  27. [27]

    Bellamy and T

    G. Bellamy and T. Schedler , title =. Lett. Math. Phys. , year =

  28. [28]

    and Ritter, A

    Benedetti, G. and Ritter, A. F. , title =. Internat. J. Math. , year =

  29. [29]

    , journal=

    Bertin, J. , journal=. 2012 , pages=

    2012

  30. [30]

    Etingof’s conjecture for quantized quiver varieties , author=. Invent. Math. , volume=. 2021 , publisher=

    2021

  31. [31]

    Bezrukavnikov, R. V. and Kaledin, D. B. , TITLE =. Tr. Mat. Inst. Steklova , FJOURNAL =. 2004 , NUMBER =

    2004

  32. [32]

    Bia. Ann. of Math. , volume=

  33. [33]

    Some properties of the decompositions of algebraic varieties determined by actions of a torus , fjournal =

    Bia. Some properties of the decompositions of algebraic varieties determined by actions of a torus , fjournal =. Bull. Acad. Pol. Sci., S. 1976 , language =

    1976

  34. [34]

    Bielawski and A

    R. Bielawski and A. Dancer , title =. Communications in Analysis and Geometry , volume=. 2000 , publisher=

    2000

  35. [35]

    Asian Journal of Mathematics , volume=

    Effective Hironaka resolution and its complexity , author=. Asian Journal of Mathematics , volume=. 2011 , publisher=

    2011

  36. [36]

    Publications of the Research Institute for Mathematical Sciences , volume=

    Functoriality in resolution of singularities , author=. Publications of the Research Institute for Mathematical Sciences , volume=

  37. [37]

    Biran and O

    P. Biran and O. Cornea , title =. 2007 , journal=

    2007

  38. [38]

    Biran and O

    P. Biran and O. Cornea , title =. Geom. Topol. , year =

  39. [39]

    Biran and O

    P. Biran and O. Cornea , title =. New perspectives and challenges in symplectic field theory , year =

  40. [40]

    arXiv preprint arXiv:0806.1050 , year=

    Irregular connections and Kac-Moody root systems , author=. arXiv preprint arXiv:0806.1050 , year=

    arXiv

  41. [41]

    Publications math

    Simply-laced isomonodromy systems , author=. Publications math

  42. [42]

    Boalch , booktitle=

    P. Boalch , booktitle=. 2018 , publisher=

    2018

  43. [43]

    , title =

    Borel, A. , title =. 1954 , doi =. https://www.pnas.org/content/40/12/1147.full.pdf , journal =

    1954

  44. [44]

    , TITLE =

    Bott, R. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1954 , PAGES =

    1954

  45. [45]

    1982 , publisher=

    Differential Forms in Algebraic Topology , author=. 1982 , publisher=

    1982

  46. [46]

    Bourgeois , title =

    F. Bourgeois , title =. 2002 , pages =

    2002

  47. [47]

    Bourgeois and A

    F. Bourgeois and A. Oancea , title =. Invent. Math. , year =

  48. [48]

    Bourgeois and A

    F. Bourgeois and A. Oancea , title =. Duke Math. J. , volume=. 2009 , publisher=

    2009

  49. [49]

    Bourgeois and T

    F. Bourgeois and T. Ekholm and Y. Eliashberg , title =. Geometry & topology , volume=. 2012 , publisher=

    2012

  50. [50]

    Bourgeois and A

    F. Bourgeois and A. Oancea , journal=. 2013 , publisher=

    2013

  51. [51]

    Bourgeois and A

    F. Bourgeois and A. Oancea , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2017 , NUMBER =

    2017

  52. [52]

    Braden and A

    T. Braden and A. Licata and N. Proudfoot and B. Webster , title =. 2016 , volume =

    2016

  53. [53]

    Braden and N

    T. Braden and N. Proudfoot and B. Webster , title =. Ast. 2016 , volume =

    2016

  54. [54]

    Brav and V

    C. Brav and V. Bussi and D. Joyce and O. Ben-Bassat , year =. A Darboux theorem for derived schemes with shifted symplectic structure , volume =. Journal of the American Mathematical Society , doi =

  55. [55]

    Braverman and D

    A. Braverman and D. and A. Okounkov , title =. Adv. Math , year =

  56. [56]

    and Finkelberg, M

    Braverman, A. and Finkelberg, M. and Nakajima, H. , title = ". Adv. Theor. Math. Phys. 2018. doi:10.4310/ATMP.2018.v22.n5.a1. arXiv:1601.03586

    work page doi:10.4310/atmp.2018.v22.n5.a1 2018

  57. [57]

    G. E. Bredon , title =. 1967 , address =

    1967

  58. [58]

    G. E. Bredon , title =. 1972 , address =

    1972

  59. [59]

    Brian. Invent. Math. , volume=. 1977 , publisher=

    1977

  60. [60]

    and King, A

    Bridgeland, T. and King, A. and Reid, M. , journal=. The

  61. [61]

    , journal=

    Brion, M. , journal=. Equivariant. 1997 , publisher=

    1997

  62. [62]

    , TITLE =

    Brion, M. , TITLE =. J. Math. Sci. Univ. Tokyo , FJOURNAL =. 2015 , NUMBER =

    2015

  63. [63]

    Brundan and V

    J. Brundan and V. Ostrik , title =. Transform. Groups , year =

  64. [64]

    Brundan and C

    J. Brundan and C. Stroppel , title =. Mosc. Math. J. , year =

  65. [65]

    Bussi , title =

    V. Bussi , title =. 2014 , journal =

    2014

  66. [66]

    , booktitle=

    Calabi, E. , booktitle=. M

  67. [67]

    Canas da Silva, A. M. L. G. , title =. Ph.D. Thesis, Massachusetts Institute of Technology , year =

  68. [68]

    Carrell and A.J

    J.B. Carrell and A.J. Sommese , title =. Comment. Math. Helv , year =

  69. [69]

    Carrell and R.M

    J.B. Carrell and R.M. Goresky , title =. Invent. Math , year =

  70. [70]

    Bigrading the symplectic Khovanov cohomology , author=. Algebr. Geom. Topol , volume=. 2023 , publisher=

    2023

  71. [71]

    Higher dimensional birational geometry , pages=

    Characterizations of projective space and applications to complex symplectic manifolds , author=. Higher dimensional birational geometry , pages=. 2002 , publisher=

    2002

  72. [72]

    Chriss and V

    N. Chriss and V. Ginzburg , title =. 1997 , address =

    1997

  73. [73]

    and Diaconescu, D.-E

    Chuang, W. and Diaconescu, D.-E. and Donagi, R. and Pantev, T. , journal=. 2015 , publisher=

    2015

  74. [74]

    Cieliebak and A

    K. Cieliebak and A. Floer and H. Hofer and K. Wysocki , title =. Math. Z. , year =

  75. [75]

    Cieliebak , title =

    K. Cieliebak , title =. J. Eur. Math. Soc , year =

  76. [76]

    Collingwood and W

    D. Collingwood and W. McGovern , title =. 1993 , address =

    1993

  77. [77]

    Conrad and O

    B. Conrad and O. Gabber and G. Prasad , title =. 2010 , address =

    2010

  78. [78]

    arXiv preprint arXiv:2406.01379 , year=

    C. arXiv preprint arXiv:2406.01379 , year=

    arXiv

  79. [79]

    2011 , pages =

    Toric varieties , author=. 2011 , pages =

    2011

  80. [80]

    Cannas da Silva , title =

    A. Cannas da Silva , title =. 2001 , address =

    2001

Showing first 80 references.