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arxiv: 2606.02903 · v1 · pith:FCCGV2TSnew · submitted 2026-06-01 · 🧮 math.AG · math-ph· math.MP

Cohomology of complex supertori

Hargun Bhatia , Soumya Ganguly , Zhiyuan Jiang , Jeffrey M Rabin , Steven V Sam This is my paper

Pith reviewed 2026-06-28 12:16 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.MP
keywords supertoricoherent cohomologyLie algebra cohomologystructure sheafPoincaré dualityPicard groupssuperspace quotients
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The pith

The coherent cohomology groups of the structure sheaf on complex supertori reduce to Lie algebra cohomology of the translation group.

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

The paper examines supertori formed as quotients of affine superspace by translations generated by algebraically independent odd parameters. It presents the ring of global sections by generators and relations. The central step reduces the coherent cohomology of the structure sheaf to a Lie algebra cohomology computation for the translation group. This reduction also preserves compatibility of Poincaré duality with the group cohomology and yields the Picard groups, illustrated by explicit tables.

Core claim

For supertori defined as quotients of affine superspace by algebraically independent odd translations, the coherent cohomology groups of the structure sheaf are completely determined by reducing the problem to Lie algebra cohomology of the translation group; the ring of global sections is described by generators and relations; Poincaré duality is compatible between sheaf and group cohomology; and the Picard groups are computed.

What carries the argument

The reduction of coherent sheaf cohomology on the supertorus to Lie algebra cohomology of its translation group.

Load-bearing premise

The algebraic independence of the odd parameters suffices for the reduction from sheaf cohomology to Lie algebra cohomology to hold without further vanishing or convergence conditions.

What would settle it

A concrete choice of algebraically independent odd parameters where the sheaf cohomology groups fail to match the Lie algebra cohomology groups would disprove the reduction.

read the original abstract

We consider supertori which are quotients of affine superspace by translations by algebraically independent odd parameters. Specifically, we describe the ring structure of its space of global sections by generators and relations and completely determine the coherent cohomology groups of its structure sheaf by reducing it to a problem of Lie algebra cohomology. We also show that Poincar\'e duality on sheaf cohomology is compatible with that of the group cohomology of the translation group and give explicit tables of examples. Finally, we compute the Picard groups of these supertori.

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 / 3 minor

Summary. The paper considers supertori formed as quotients of affine superspace by translations with algebraically independent odd parameters. It describes the ring structure on global sections via generators and relations, determines the coherent cohomology groups of the structure sheaf by reducing the computation to Lie algebra cohomology of the translation group, establishes compatibility of Poincaré duality between sheaf and group cohomology, supplies explicit tables of examples, and computes the Picard groups of these supertori.

Significance. If the central reduction holds, the work furnishes a complete, explicit determination of the cohomology of the structure sheaf on these supertori together with duality compatibility and Picard-group computations. The provision of explicit tables of examples and the compatibility statement between sheaf and group cohomology constitute concrete strengths that make the results directly usable for further calculations in supergeometry.

minor comments (3)
  1. [§2] §2, paragraph following Definition 2.3: the precise statement of algebraic independence of the odd parameters is used to justify the reduction but is not restated when the Lie-algebra cohomology computation is invoked; a one-sentence reminder would improve readability.
  2. [Table 4] Table 4 (period-3 row): the displayed basis elements for H^1 are given without indicating the grading or the action of the odd parameters; adding this information would make the table self-contained.
  3. [§5] The final section on Picard groups cites the cohomology computation but does not explicitly record which vanishing results are used; a short sentence linking back to the relevant cohomology table would clarify the argument.

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 major comments were provided in the report, so we have no specific points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The central claim reduces coherent sheaf cohomology on the supertorus to Lie algebra cohomology of the translation group as an external standard computation under algebraic independence of odd parameters. This reduction is not self-definitional, does not rename a fitted input as a prediction, and relies on no load-bearing self-citation chain. Explicit ring structure, duality compatibility, and example tables are provided independently. The derivation remains self-contained against external benchmarks with no reduction of the target result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard facts from algebraic geometry and Lie algebra cohomology; no free parameters or new entities are introduced in the abstract. The key step is a reduction whose validity is taken as a domain assumption.

axioms (1)
  • domain assumption Coherent sheaf cohomology on the quotient can be reduced to Lie algebra cohomology of the translation group when the odd parameters are algebraically independent.
    Stated as the method used to determine the cohomology groups (abstract).

pith-pipeline@v0.9.1-grok · 5620 in / 1260 out tokens · 20069 ms · 2026-06-28T12:16:17.889696+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

40 extracted references · 18 canonical work pages

  1. [1]

    2026 , eprint=

    Poincar\'e Duality and Supergravity , author=. 2026 , eprint=

    2026

  2. [2]

    Shabat, B. V. , TITLE =. 1992 , PAGES =. doi:10.1090/mmono/110 , URL =

    work page doi:10.1090/mmono/110 1992

  3. [3]

    1979 , PAGES =

    Grauert, Hans and Remmert, Reinhold , TITLE =. 1979 , PAGES =

    1979

  4. [4]

    , TITLE =

    Stanley, Richard P. , TITLE =. 2024 , PAGES =

    2024

  5. [5]

    , TITLE =

    Humphreys, James E. , TITLE =. 1972 , PAGES =

    1972

  6. [6]

    , TITLE =

    Sam, Steven V. , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s00029-026-01131-3 , URL =

    work page doi:10.1007/s00029-026-01131-3 2026

  7. [7]

    Sam, Steven V and Snowden, Andrew , TITLE =

  8. [8]

    Sam, Steven V and VandeBogert, Keller and Weyman, Jerzy , TITLE =

  9. [9]

    Sam, Steven V and Snowden, Andrew , TITLE=

  10. [10]

    Sam, Steven V and Snowden, Andrew , TITLE =. Algebr. Geom. , FJOURNAL =. 2024 , NUMBER =. doi:10.14231/ag-2024-002 , URL =

    work page doi:10.14231/ag-2024-002 2024

  11. [11]

    Cohomology of

    Weyman, Jerzy , TITLE =. 2003 , PAGES =. doi:10.1017/CBO9780511546556 , URL =

    work page doi:10.1017/cbo9780511546556 2003

  12. [12]

    1991 , PAGES =

    Fulton, William and Harris, Joe , TITLE =. 1991 , PAGES =. doi:10.1007/978-1-4612-0979-9 , URL =

    work page doi:10.1007/978-1-4612-0979-9 1991

  13. [13]

    1978 , PAGES =

    Griffiths, Phillip and Harris, Joseph , TITLE =. 1978 , PAGES =

    1978

  14. [14]

    Bergvelt, M. J. and Rabin, J. M. , TITLE =. Duke Math. J. , FJOURNAL =. 1999 , NUMBER =. doi:10.1215/S0012-7094-99-09801-0 , URL =

    work page doi:10.1215/s0012-7094-99-09801-0 1999

  15. [15]

    2020 , eprint=

    An introduction to supermanifolds and supersymmetry , author=. 2020 , eprint=

    2020

  16. [16]

    , TITLE =

    Manin, Yuri I. , TITLE =. 1997 , PAGES =. doi:10.1007/978-3-662-07386-5 , URL =

    work page doi:10.1007/978-3-662-07386-5 1997

  17. [17]

    Pure Appl

    Witten, Edward , TITLE =. Pure Appl. Math. Q. , FJOURNAL =. 2019 , NUMBER =. doi:10.4310/PAMQ.2019.v15.n1.a1 , URL =

    work page doi:10.4310/pamq.2019.v15.n1.a1 2019

  18. [18]

    1970 , PAGES =

    Mumford, David , TITLE =. 1970 , PAGES =

    1970

  19. [19]

    Cambridge (1994) DOI: 10.1017/CBO9781139644136 Universidad Nacional de Colombia - Sede Bogotá Current address: Campus Universitario Email address:mareyesv@unal.edu.co

    Weibel, Charles A. , TITLE =. 1994 , PAGES =. doi:10.1017/CBO9781139644136 , URL =

    work page doi:10.1017/cbo9781139644136 1994

  20. [20]

    Haske, Carl and Wells, Jr., R. O. , TITLE =. Duke Math. J. , FJOURNAL =. 1987 , NUMBER =. doi:10.1215/S0012-7094-87-05421-4 , URL =

    work page doi:10.1215/s0012-7094-87-05421-4 1987

  21. [21]

    LeBrun, Claude and Poon, Yat Sun and Wells, Jr., R. O. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1990 , NUMBER =

    1990

  22. [22]

    Geometric methods in physics

    Voronov, Theodore , TITLE =. Geometric methods in physics. 2018 , ISBN =

    2018

  23. [23]

    Rabin and Brendon Rhoades , keywords =

    Jesse Kim and Jeffrey M. Rabin and Brendon Rhoades , keywords =. The combinatorics of supertorus sheaf cohomology , journal =. 2023 , issn =. doi:https://doi.org/10.1016/j.geomphys.2023.104963 , url =

    work page doi:10.1016/j.geomphys.2023.104963 2023

  24. [24]

    Differential Geom

    Noja, Simone , TITLE =. Differential Geom. Appl. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.difgeo.2023.101999 , URL =

    work page doi:10.1016/j.difgeo.2023.101999 2023

  25. [25]

    Hilbert and Picard superschemes , author=

    Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes , author=. 2023 , eprint=

    2023

  26. [26]

    1990 , PAGES =

    H\"ormander, Lars , TITLE =. 1990 , PAGES =

    1990

  27. [27]

    1977 , PAGES =

    Hartshorne, Robin , TITLE =. 1977 , PAGES =

    1977

  28. [28]

    , TITLE =

    Bott, Raoul and Tu, Loring W. , TITLE =. 1982 , PAGES =

    1982

  29. [29]

    2000 , PAGES =

    Neukirch, J\"urgen and Schmidt, Alexander and Wingberg, Kay , TITLE =. 2000 , PAGES =

    2000

  30. [30]

    1995 , PAGES =

    Mac Lane, Saunders , TITLE =. 1995 , PAGES =

    1995

  31. [31]

    , TITLE =

    Rabin, Jeffrey M. , TITLE =. J. Geom. Phys. , FJOURNAL =. 1995 , NUMBER =. doi:10.1016/0393-0440(94)00012-S , URL =

    work page doi:10.1016/0393-0440(94)00012-s 1995

  32. [32]

    Bartocci, Claudio and Bruzzo, Ugo and Hern\'andez Ruip\'erez, Daniel and Pestov, Vladimir , TITLE =. Bull. Austral. Math. Soc. , FJOURNAL =. 1998 , NUMBER =. doi:10.1017/S0004972700032044 , URL =

    work page doi:10.1017/s0004972700032044 1998

  33. [33]

    2018 , note=

    TOPICS IN ALGEBRAIC SUPERGEOMETRY OVER PROJECTIVE SPACES , author=. 2018 , note=

    2018

  34. [34]

    Alldridge, Alexander and Hilgert, Joachim , TITLE =. J. Lie Theory , FJOURNAL =. 2010 , NUMBER =

    2010

  35. [35]

    , TITLE =

    Wachs, Michelle L. , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 1985 , NUMBER =. doi:10.1016/0097-3165(85)90091-3 , URL =

    work page doi:10.1016/0097-3165(85)90091-3 1985

  36. [36]

    Chen, William Y. C. and Krattenthaler, Christian and Yang, Arthur L. B. , TITLE =. Graphs Combin. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00373-004-0593-9 , URL =

    work page doi:10.1007/s00373-004-0593-9 2005

  37. [37]

    1997 , PAGES =

    Fulton, William , TITLE =. 1997 , PAGES =

    1997

  38. [38]

    Macdonald, I. G. , TITLE =. 1995 , PAGES =

    1995

  39. [39]

    , TITLE =

    Gessel, Ira M. , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 1990 , NUMBER =. doi:10.1016/0097-3165(90)90060-A , URL =

    work page doi:10.1016/0097-3165(90)90060-a 1990

  40. [40]

    Bruzzo, Ugo and Hern\'andez Ruip\'erez, Daniel and Polishchuk, Alexander , TITLE =. Adv. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.aim.2023.108890 , URL =

    work page doi:10.1016/j.aim.2023.108890 2023