pith. sign in

arxiv: 2607.01355 · v1 · pith:SXLBQFCLnew · submitted 2026-07-01 · 🧮 math.AG · math.RT

Hecke operators on symplectic surfaces and chi-independence

Ben Davison , Lucien Hennecart , Tasuki Kinjo , Olivier Schiffmann , Eric Vasserot This is my paper

Pith reviewed 2026-07-03 18:35 UTC · model grok-4.3

classification 🧮 math.AG math.RT
keywords BPS cohomologychi-independenceHecke operatorscohomological Hall algebrasymplectic surfacesmoduli spaces of sheavestautological classesMukai vectors
0
0 comments X

The pith

BPS cohomology of one-dimensional sheaves on quasi-projective symplectic surfaces is chi-independent relative to the Chow variety.

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

The paper proves Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces. It constructs this result using Hecke operators that act on BPS cohomology by modifying sheaves with zero-dimensional quotients, derived from a bialgebra structure on the cohomological Hall algebra. A sympathetic reader would care because the result resolves an open conjecture and simultaneously identifies the associated BPS Lie algebra with the subspace of tautological classes, extending a known generation theorem. The argument relies on reducing a factorization coproduct from three-dimensional algebras to obtain the necessary bialgebra structure.

Core claim

We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise t

What carries the argument

Hecke operators on BPS cohomology that modify one-dimensional sheaves by zero-dimensional quotients, obtained from the bialgebra structure on the cohomological Hall algebra via dimensional reduction from a 3d factorization coproduct.

If this is right

  • BPS cohomology of the moduli spaces is independent of the Euler characteristic relative to the Chow variety.
  • The BPS Lie algebra for one-dimensional Mukai vectors coincides exactly with the subspace of tautological classes.
  • Markman's tautological generation theorem extends from primitive to arbitrary Mukai vectors.
  • A global BPS Lie algebra is attached to the stack of all coherent sheaves on the symplectic variety S.

Where Pith is reading between the lines

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

  • The same bialgebra and Hecke construction may yield explicit formulas for BPS invariants in low-dimensional examples.
  • The approach could extend to compute similar invariants for moduli spaces with different stability conditions.
  • The identification with tautological classes suggests these classes control the full ring structure in the BPS setting.

Load-bearing premise

The bialgebra structure on the cohomological Hall algebra arises by dimensional reduction, and the affinized BPS cohomology of the semistable locus equals the primitive part of the coproduct on the full moduli stack.

What would settle it

Compute the graded dimensions of the BPS cohomology for two different Euler characteristics on the same component of the Chow variety for a fixed symplectic surface and one-dimensional Mukai vector; a mismatch would falsify chi-independence.

read the original abstract

We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise to a global BPS Lie algebra attached to the stack of coherent sheaves on S. The link between this structure and the applications to chi-independence and tautological generation is provided by Hecke operators on BPS cohomology, which modify one-dimensional sheaves by zero-dimensional quotients. To make this construction work, we prove that there is an identification between the affinized BPS cohomology of the semistable locus and the primitive part of the coproduct on the entire moduli stack

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 proves Toda's χ-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. It also identifies the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, extending Markman's tautological generation theorem. The central technical input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S, obtained by dimensional reduction from a factorization coproduct for 3d cohomological Hall algebras; this yields a global BPS Lie algebra. The link to the applications is provided by Hecke operators on BPS cohomology together with the identification of the affinized BPS cohomology of the semistable locus with the primitive part of the coproduct on the full moduli stack.

Significance. If the derivations hold, the work constitutes a substantial advance on chi-independence and tautological generation for BPS cohomology in the setting of symplectic surfaces. The construction of the bialgebra via dimensional reduction supplies a concrete structural input that is then applied via Hecke operators, and the identification with tautological classes extends an existing theorem from primitive to arbitrary Mukai vectors. These results strengthen the dictionary between cohomological Hall algebras, BPS Lie algebras, and geometric invariants on moduli stacks.

minor comments (2)
  1. §2.3: the definition of the affinized BPS cohomology is introduced after its first use in the statement of the key identification; moving the definition forward would improve readability.
  2. Notation for the Mukai vector v and the Chow variety is used consistently but would benefit from a single consolidated table of symbols in the preliminaries.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were listed in the report, so we have no specific points requiring response or revision at this stage. We are prepared to incorporate any minor suggestions that may arise.

Circularity Check

0 steps flagged

Derivation is self-contained with independent structure inputs

full rationale

The paper constructs a bialgebra on the CoHA via dimensional reduction from a 3d factorization coproduct, states an identification of affinized BPS cohomology on the semistable locus with the primitive part of the coproduct as a theorem with supporting arguments, and applies Hecke operators to reach chi-independence and tautological generation. These steps are presented with explicit definitions and reductions internal to the manuscript; no load-bearing step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain. The derivation chain remains independent of its target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities are identifiable. The work relies on established structures like cohomological Hall algebras and BPS cohomology.

pith-pipeline@v0.9.1-grok · 5740 in / 1314 out tokens · 43709 ms · 2026-07-03T18:35:19.368616+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

94 extracted references · 24 canonical work pages · 4 internal anchors

  1. [1]

    Good Moduli Spaces in Derived Algebraic Geometry

    E. Ahlqvist, J. Hekking, M. Pernice, M. Savvas,Good moduli spaces in derived algebraic geometry, arXiv:2309.16574 (2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  2. [2]

    Alper, D

    J. Alper, D. Halpern-Leistner, J. Heinloth,Existence of moduli spaces for algebraic stacks, Invent. Math.234(2023), 949–1038

    2023

  3. [3]

    Bayer, E

    A. Bayer, E. Macr` ı,MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math.198(2014), 505–590

    2014

  4. [4]

    Beauville, M

    A. Beauville, M. Narasimhan, S. Ramanan,Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398(1989), 169–179

    1989

  5. [5]

    A. A. Beilinson, J. Bernstein, P. Deligne, O. Gabber,Faisceaux pervers, Ast´ erisque,100, Soci´ et´ e Math´ ematique de France, Paris, 1982, 5–171

    1982

  6. [6]

    Birkenhake, H

    C. Birkenhake, H. Lange,Complex Abelian Varieties, Grundlehren Math. Wiss.,302, Springer Verlag, (1992)

    1992

  7. [7]

    Botta, B

    T. Botta, B. Davison,Okounkov’s conjecture via BPS Lie algebras, arXiv:2312.14008 (2023)

    work page arXiv 2023

  8. [8]

    Bottini,Stable sheaves on K3 surfaces via wall-crossing, Kyoto J

    A. Bottini,Stable sheaves on K3 surfaces via wall-crossing, Kyoto J. Math.64(2024), no. 2, 459–499

    2024

  9. [9]

    Bozec, D

    T. Bozec, D. Calaque, S. Scherotzke,Relative critical loci and quiver moduli, Ann. Sci. ´Ec. Norm. Sup´ er. (4)57(2024), no. 2, 553–614

    2024

  10. [10]

    C. Brav, V. Bussi, D. Dupont, D. Joyce, B. Szendroi,Symmetries and stabilization for sheaves of vanishing cycles, with an appendix by J¨ org Sch¨ urmann, J. Singul.11(2015) 85–151

    2015

  11. [11]

    C. Brav, T. Dyckerhoff,Relative Calabi-Yau structures, Compos. Math.155(2019), 372–412

    2019

  12. [12]

    C. Brav, T. Dyckerhoff,Relative Calabi-Yau structures II: shifted Lagrangians in the moduli of objects, Selecta Math. (N.S.)27(2021)

    2021

  13. [13]

    Bridgeland,Fourier–Mukai transforms for elliptic surfaces, J

    T. Bridgeland,Fourier–Mukai transforms for elliptic surfaces, J. Reine Angew. Math.498(1998), 115–133

    1998

  14. [14]

    C. Bu, B. Davison, A. Ib´ a˜ nez N´ u˜ nez, T. Kinjo, T. P˘ adurariu,Cohomology of symmetric stacks, arXiv:2502.04253, (2025)

    work page arXiv 2025

  15. [15]

    C. Bu, D. Halpern-Leistner, A. Ib´ a˜ nez N´ u˜ nez, T. Kinjo,Intrinsic Donaldson–Thomas theory. I. Component lattices of stacks, arXiv:2502.13892, (2025). HECKE OPERATORS ON SYMPLECTIC SURFACES ANDχ-INDEPENDENCE 91

    work page arXiv 2025

  16. [16]

    Calaque,Lagrangian structures on mapping stacks and semi-classical TFTs, Contemp

    D. Calaque,Lagrangian structures on mapping stacks and semi-classical TFTs, Contemp. Math.643(2015), 1–23

    2015

  17. [17]

    Calaque,Shifted cotangent stacks are shifted symplectic, Ann

    D. Calaque,Shifted cotangent stacks are shifted symplectic, Ann. Fac. Sci. Toulouse Math. (6) 28 (2019), 67–90

    2019

  18. [18]

    Calaque, P

    D. Calaque, P. Safronov,Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone, arXiv:2407.08622, (2024)

    work page arXiv 2024

  19. [19]

    Costello, O

    K. Costello, O. Gwilliam,Factorization algebras in quantum field theory, Vol. 1, New Mathematical Monographs31, Cambridge Univ. Press, Cambridge, (2017)

    2017

  20. [20]

    Davison,The critical CoHA of a quiver with potential, Q

    B. Davison,The critical CoHA of a quiver with potential, Q. J. Math.68(2017), no. 2, 635–703

    2017

  21. [21]

    Davison, S

    B. Davison, S. Meinhardt,Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras, Invent. Math.221(2020), 777–871

    2020

  22. [22]

    Davison,The integrality conjecture and the cohomology of preprojective stacks, J

    B. Davison,The integrality conjecture and the cohomology of preprojective stacks, J. Reine Angew. Math.804(2023), 105–154

    2023

  23. [23]

    Davison,Purity and 2-Calabi-Yau categories, Invent

    B. Davison,Purity and 2-Calabi-Yau categories, Invent. Math.238(2024), no. 1, 69–173

    2024

  24. [24]

    Davison,Affine BPS algebras, W algebras, and the cohomological Hall algebra ofA 2, inHomotopical methods in Geometry and Physics, of Contemp

    B. Davison,Affine BPS algebras, W algebras, and the cohomological Hall algebra ofA 2, inHomotopical methods in Geometry and Physics, of Contemp. Math., 841, Amer. Math. Soc., Providence (2026), 163–199

    2026

  25. [25]

    Davison, L

    B. Davison, L. Hennecart, S. Schlegel Mejia,BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks, arXiv:2212.07668, (2022)

    work page arXiv 2022

  26. [26]

    Davison, L

    B. Davison, L. Hennecart, S. Schlegel Mejia,BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories, arXiv:2303.12592, (2023)

    work page arXiv 2023

  27. [27]

    Descombes,Hyperbolic localization in Donaldson–Thomas theory, arXiv:2506.22400, (2025)

    P. Descombes,Hyperbolic localization in Donaldson–Thomas theory, arXiv:2506.22400, (2025)

    work page arXiv 2025

  28. [28]

    Nilpotent cohomological Hall algebras of surfaces

    D.-E. Diaconescu, M. Porta, F. Sala, O. Schiffmann, E. Vasserot,Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians, arXiv:2502.19445, (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  29. [29]

    Gaitsgory, J

    D. Gaitsgory, J. Lurie,Weil’s conjecture for function fields. Vol. 1, Ann. of Math. Stud.,199Princeton University Press, Princeton, NJ, 2019

    2019

  30. [30]

    G¨ ottsche, W

    L. G¨ ottsche, W. Soergel,Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann.296(1993), no. 2, 235–245

    1993

  31. [31]

    Groechenig, D

    M. Groechenig, D. Wyss, P. Ziegler,χ-independence for K3-surfaces viap-adic integration, (2026)

    2026

  32. [32]

    Halpern-Leistner,On the structure of instability in moduli theory, arXiv:1411.0627, (2014)

    D. Halpern-Leistner,On the structure of instability in moduli theory, arXiv:1411.0627, (2014)

    work page arXiv 2014

  33. [33]

    Theta-stratifications, Theta-reductive stacks, and applications

    D. Halpern-Leistner,Theta-stratifications, Theta-reductive stacks, and applications, arXiv:1608.04797, (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  34. [34]

    Hausel, M

    T. Hausel, M. Thaddeus,Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math.153(2003), no. 1, 197–229

    2003

  35. [35]

    Hennecart, T

    L. Hennecart, T. Kinjo,The BPS decomposition theorem, arXiv:2509.21298, (2025)

    work page arXiv 2025

  36. [36]

    Holstein, M

    J. Holstein, M. Porta,Analytification of mapping stacks, Algebr. Geom.12(2025), 13–83

    2025

  37. [37]

    Huybrechts,Compact hyper-K¨ ahler manifolds: basic results, Invent

    D. Huybrechts,Compact hyper-K¨ ahler manifolds: basic results, Invent. Math.135(1999), no. 1, 63–113

    1999

  38. [38]

    Huybrechts,Lectures onK3surfaces, Cambridge Stud

    D. Huybrechts,Lectures onK3surfaces, Cambridge Stud. Adv. Math.158Cambridge University Press, Cambridge, (2016)

    2016

  39. [39]

    Huybrechts, M

    D. Huybrechts, M. Lehn,The geometry of moduli spaces of sheaves, Aspects Math. E31 Friedr. Vieweg & Sohn, Braun- schweig, (1997)

    1997

  40. [40]

    Ib´ anez N´ unez,Refined Harder-Narasimhan filtrations in moduli theory, arXiv:2311.18050, (2023)

    A. Ib´ anez N´ unez,Refined Harder-Narasimhan filtrations in moduli theory, arXiv:2311.18050, (2023)

    work page arXiv 2023

  41. [41]

    Jindal,CoHA of cyclic quivers and an integral form of affine Yangians, arXiv:2408.02618, (2024)

    S. Jindal,CoHA of cyclic quivers and an integral form of affine Yangians, arXiv:2408.02618, (2024)

    work page arXiv 2024

  42. [42]

    Jindal, S

    S. Jindal, S. Kaubrys, A. Latyntsev,Critical CoHAs, vertex coalgebras and deformed Drinfeld coproducts, arXiv:2603.21707, (2026)

    work page arXiv 2026

  43. [43]

    Joyce, P

    D. Joyce, P. Safronov,A Lagrangian neighbourhood theorem for shifted symplectic derived schemes, Ann. Fac. Sci. Toulouse Math. (6) 28 (2019), no. 5, 831–908

    2019

  44. [44]

    Kaledin, M

    D. Kaledin, M. Lehn, C. Sorger,Singular symplectic moduli spaces, Invent. Math.164(2006), 591–614

    2006

  45. [45]

    Kapranov, E

    M. Kapranov, E. Vasserot,The cohomological Hall algebra of a surface and factorization cohomology, J. Eur. Math. Soc. (JEMS)25(2023), 4221–4289

    2023

  46. [46]

    Kashiwara, P

    M. Kashiwara, P. Schapira,Categories and sheaves, Grundlehren Math. Wiss.,352Springer-Verlag, Berlin, (2006)

    2006

  47. [47]

    Kashiwara, P

    M. Kashiwara, P. Schapira,Sheaves on Manifolds, Grundlehren Math. Wiss.,292Springer-Verlag, Berlin, (1990)

    1990

  48. [48]

    Keller,Deformed Calabi-Yau completions, J

    B. Keller,Deformed Calabi-Yau completions, J. Reine Angew. Math.654(2011), 125–180

    2011

  49. [49]

    A. A. Khan,Virtual fundamental classes of derived stacks I, arXiv:1909.01332, (2019)

    work page arXiv 1909

  50. [50]

    A. A. Khan, C. Ravi,Generalized cohomology theories for algebraic stacks, Adv. Math.458(2024), article no. 109975, 104 pp. 92 BEN DAVISON, LUCIEN HENNECART, TASUKI KINJO, OLIVIER SCHIFFMANN AND ERIC VASSEROT

    2024

  51. [51]

    A. A. Khan, T. Kinjo,3D cohomological Hall algebras for local surfaces,https://www.preschema.com/papers/ dimredcoha.pdf

  52. [52]

    A. A. Khan, T. Kinjo, H. Park, P. Safronov,Shifted Lagrangian classes, to appear

  53. [53]

    A. A. Khan, T. Kinjo, H. Park, P. Safronov,Perverse pullbacks, arXiv:2510.16563 (2025)

    work page arXiv 2025

  54. [54]

    Kinjo,Dimensional reduction in cohomological Donaldson–Thomas theory, Compos

    T. Kinjo,Dimensional reduction in cohomological Donaldson–Thomas theory, Compos. Math.158(2022), 123–167

    2022

  55. [55]

    Kinjo,Virtual classes via vanishing cycles, arXiv:2109.06468, (2021)

    T. Kinjo,Virtual classes via vanishing cycles, arXiv:2109.06468, (2021)

    work page arXiv 2021

  56. [56]

    Kinjo, N

    T. Kinjo, N. Koseki,Cohomologicalχ-independence for Higgs bundles and Gopakumar–Vafa invariants, J. Eur. Math. Soc. (JEMS)28(2026), no. 2, 619–671

    2026

  57. [57]

    Kinjo, H

    T. Kinjo, H. Park, P. Safronov,Cohomological Hall algebras for 3-Calabi-Yau categories, arXiv:2406.12838 (2024)

    work page arXiv 2024

  58. [58]

    Lurie,Higher Algebra, available athttps://www.math.ias.edu/ ~lurie/(2017)

    J. Lurie,Higher Algebra, available athttps://www.math.ias.edu/ ~lurie/(2017)

    2017

  59. [59]

    Lurie,Higher Topos Theory, Ann

    J. Lurie,Higher Topos Theory, Ann. of Math. Stud.,170, Princeton University Press, Princeton, NJ, (2009)

    2009

  60. [60]

    Lurie,Derived Algebraic Geometry XIII: Rational andp-adic Homotopy Theory, (2011)

    J. Lurie,Derived Algebraic Geometry XIII: Rational andp-adic Homotopy Theory, (2011)

    2011

  61. [61]

    Markman,Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces, J

    E. Markman,Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces, J. Reine Angew. Math.544(2002), 61–82

    2002

  62. [62]

    Maulik, J

    D. Maulik, J. Shen,TheP=Wconjecture forGL n, Ann. of Math. (2)200(2024), no. 2, 529–556

    2024

  63. [63]

    Maulik, R

    D. Maulik, R. Thomas,Sheaf counting on local K3 surfaces, Pure Appl. Math. Q.14(2018), no. 3-4, 419–441

    2018

  64. [64]

    Maulik, Y

    D. Maulik, Y. Toda,Gopakumar–Vafa invariants via vanishing cycles, Invent. Math.213(2018), 1017–1097

    2018

  65. [65]

    Mellit, A

    A. Mellit, A. Minets, O. Schiffmann, T. Hausel,P=WviaH 2, arXiv:2209.05429 (2022)

    work page arXiv 2022

  66. [66]

    Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras

    A. Mellit, A. Minets, O. Schiffmann, E. Vasserot,Coherent sheaves on surfaces, COHAs and deformedW 1+∞-algebras, arXiv:2311.13415 (2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  67. [67]

    Milnor, J

    J. Milnor, J. Moore,On the structure of Hopf algebras, Ann. of Math. (2)81(1965), 211–264

    1965

  68. [68]

    Minets,Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces, Selecta Math

    A. Minets,Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces, Selecta Math. (N.S.)26(2020), no. 2, Paper No. 30, 67 pp

    2020

  69. [69]

    Mukai,Symplectic structure of the moduli space of sheaves on an Abelian orK3surface, Invent

    S. Mukai,Symplectic structure of the moduli space of sheaves on an Abelian orK3surface, Invent. Math.77(1984), no. 1, 101–116

    1984

  70. [70]

    Mukai,On the moduli space of bundles on K3 surfaces

    S. Mukai,On the moduli space of bundles on K3 surfaces. I., inVector bundles on algebraic varieties (Bombay, 1984), volume 11 of Tata Inst. Fund. Res. Stud. Math., pages 341–413. Bombay, 1987

    1984

  71. [71]

    Nitsure,Schematic Harder-Narasimhan stratification, Internat

    N. Nitsure,Schematic Harder-Narasimhan stratification, Internat. J. Math.22(2011), 1365–1373

    2011

  72. [72]

    O’Grady,Desingularized moduli spaces of sheaves on aK3, J

    K. O’Grady,Desingularized moduli spaces of sheaves on aK3, J. Reine Angew. Math.512(1999), 49–117

    1999

  73. [73]

    Orlov,Equivalences of derived categories and K3 surfaces, J

    D. Orlov,Equivalences of derived categories and K3 surfaces, J. Math. Sci. (N.Y.)84(1997), no. 5, 1361–1381

    1997

  74. [74]

    Pantev, B

    T. Pantev, B. To¨ en, M. Vaqui´ e, G. Vezzosi,Shifted symplectic structures, Publ. Math. Inst. Hautes ´Etudes Sci.117 (2013), 271–328

    2013

  75. [75]

    Porta,GAGA theorems in derived complex geometry, J

    M. Porta,GAGA theorems in derived complex geometry, J. Algebraic Geom.28, no. 3, 519–565 (2019)

    2019

  76. [76]

    Porta, F

    M. Porta, F. Sala,Two dimensional categorified Hall algebras, J. Eur. Math. Soc. (JEMS)25(2023), no. 3, 1113–1205

    2023

  77. [77]

    Rozenblyum,Connections on moduli spaces and infinitesimal Hecke modifications, arXiv:2108.07745 (2021)

    N. Rozenblyum,Connections on moduli spaces and infinitesimal Hecke modifications, arXiv:2108.07745 (2021)

    work page arXiv 2021

  78. [78]

    Rydh, Families of cycles,https://people.kth.se/ ~dary/thesis/thesis-paperIV.pdf

    D. Rydh, Families of cycles,https://people.kth.se/ ~dary/thesis/thesis-paperIV.pdf

  79. [79]

    Saint-Donat,Projective models ofK3surfaces, Amer

    B. Saint-Donat,Projective models ofK3surfaces, Amer. J. Math.96(1974), 602–639

    1974

  80. [80]

    F. Sala, O. Schiffmann,Cohomological Hall algebra of Higgs sheaves on a curve, Algebr. Geom.7(2020), 346–376

    2020

Showing first 80 references.