pith. sign in

arxiv: 2212.11836 · v5 · pith:WXXMOWRTnew · submitted 2022-12-22 · 🧮 math.AG · math.AT

Spectrum of equivariant cohomology as a fixed point scheme

Tam\'as Hausel , Kamil Rychlewicz This is my paper

Pith reviewed 2026-05-24 10:25 UTC · model grok-4.3

classification 🧮 math.AG math.AT
keywords equivariant cohomologyfixed point schemereductive group actionregular actionGKM spacesflag varietiesSchubert varietiestoric varieties
0
0 comments X

The pith

The G-equivariant cohomology ring of a smooth projective variety with regular reductive group action is the coordinate ring of a regular fixed point scheme.

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

The paper shows that when a complex reductive group G acts regularly on a smooth projective variety X, the G-equivariant cohomology ring of X is isomorphic to the coordinate ring of a certain regular fixed point scheme. Regularity means every regular unipotent element of G fixes only finitely many points of X. This identification holds for partial flag varieties, smooth Schubert varieties, and Bott-Samelson varieties. A generalized version of the fixed point scheme extends the isomorphism to GKM spaces such as toric varieties.

Core claim

When a complex reductive group G acts regularly on a smooth projective variety X, the complex G-equivariant cohomology ring of X is isomorphic to the coordinate ring of a certain regular fixed point scheme.

What carries the argument

The regular fixed point scheme, a geometric object whose coordinate ring realizes the equivariant cohomology under the regularity condition on the G-action.

If this is right

  • The isomorphism holds for partial flag varieties.
  • The isomorphism holds for smooth Schubert varieties.
  • The isomorphism holds for Bott-Samelson varieties.
  • A generalized fixed point scheme yields the same type of isomorphism for GKM spaces such as toric varieties.

Where Pith is reading between the lines

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

  • The fixed point scheme may supply a uniform algebraic way to compute equivariant cohomology rings that were previously described only by generators and relations.
  • The construction could extend to other cohomology theories or to actions that are not fully regular but satisfy weaker finiteness conditions.
  • The result links equivariant cohomology directly to the geometry of schemes fixed by unipotent elements, suggesting new questions about the scheme's singularities or its relation to the variety X.

Load-bearing premise

The action of the complex reductive group G on X must be regular so that all regular unipotent elements act with finitely many fixed points.

What would settle it

For a concrete partial flag variety, explicitly compute both the G-equivariant cohomology ring and the coordinate ring of the associated regular fixed point scheme and check whether the two rings fail to be isomorphic.

Figures

Figures reproduced from arXiv: 2212.11836 by Kamil Rychlewicz, Tam\'as Hausel.

Figure 1
Figure 1. Figure 1: Spec H˚ Cˆ pP 3 q. Example 3.20. We continue Example 2.42 which already appears in [16]. The point o “ r1 : 0 : ¨ ¨ ¨ : 0s is the unique zero of e. If rz0 : z1 : ¨ ¨ ¨ : zns are the homogeneous coordinates of P n, then the scheme Z lies completely in the affine chart Xo of o, with affine coordinates xi “ zi{z0, for i “ 1, 2, . . . , n. We have Vh|x1,...,xn “ p´2x1, ´4x2, . . . , ´2nxnq and Ve|x1,...,xn “ p… view at source ↗
Figure 2
Figure 2. Figure 2: Two different views of Spec H˚ Cˆ pGrp2, 4qq. Note that all the components project bijectively to the v axis. Example 3.23. As we have defined an action of SL2pCq on any C n, we can use this to define actions on partial or full flag varieties. Let us consider the action of the upper Borel subgroup of SL2 on C 4 and the induced action on the Grassmannian Grp2, 4q of two-planes in C 2 . We can identify it wi… view at source ↗
Figure 3
Figure 3. Figure 3: Spec H˚ T pP 2 q. Example 3.24. Let us now switch to groups of higher rank. As in Example 2.41, we can consider the regular nilpotent e “ ¨ ˚˚˚˚˚˚˚˚˚˚˝ 0 1 0 0 . . . 0 0 0 1 0 . . . 0 0 0 0 1 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 0 0 . . . 1 0 0 0 0 . . . 0 ˛ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹‚ . in SLn`1. We have the regular action of SLn`1 on P n, which in particular restricts to a regular action of its upper… view at source ↗
Figure 4
Figure 4. Figure 4: Spec H˚ SL2pCq pP 4 q and Spec H˚ SL2pCq pP 5 q. Example 4.6. We continue Example 3.20. There, we found the C ˆ-equivariant cohomology of P n. Now, using the tools above, we can also find Spec HSL2pCqpP nq. We know that the map pv, xq ÞÑ pv,pI ` vfqxq maps the zeros of Ve`vh isomorphically to the zeros of Ve`v 2f . The former form the subscheme cut out by x1px1 ` 2vqpx1 ` 4vq. . .px1 ` 2nvq “ 0 in the pv, … view at source ↗
Figure 5
Figure 5. Figure 5: Two different views of Spec H˚ SL2pCq pGrp2, 4qq. Example 4.7. We continue Example 3.23. The principal SL2pCq Ă SL4pCq subgroup acts on Grp2, 4q. One can check that Vf |x1,y1,x2,y2 “ p´3y1, 4, 3x1 ´ 3y2, 3y1q. Then Ve`tf |x1,y1,x2,y2 “ px2 ´ x1y1 ´ 3ty1, ´x1 ´ y 2 1 ` y2 ` 4t, ´x1y2 ` 3tx1 ´ 3ty2, ´x2 ´ y1y2 ` 3ty1q. As before, from the first two equations of Ve`tf “ 0, we can determine x2 and y2, so Spec … view at source ↗
Figure 6
Figure 6. Figure 6: Spec H˚ SL3pCq pP 2 q. On the right the subscheme Spec H˚ SL2pCq pP 2 q is marked. Compare with [PITH_FULL_IMAGE:figures/full_fig_p046_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Affine parts of the total zero scheme for the action of B [PITH_FULL_IMAGE:figures/full_fig_p053_7.png] view at source ↗
read the original abstract

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology ring of $X$ is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties.

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 claims that if a complex reductive group G acts regularly on a smooth projective variety X (i.e., every regular unipotent element of G has finitely many fixed points), then the G-equivariant cohomology ring H_G^*(X; ℂ) is isomorphic to the coordinate ring of a certain regular fixed point scheme constructed from those fixed points. The result is illustrated on partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties, and a more general version of the scheme is shown to recover the equivariant cohomology of GKM spaces such as toric varieties.

Significance. If the stated isomorphism holds, the work supplies an explicit algebraic-geometric model for equivariant cohomology rings in terms of functions on a scheme whose points are fixed by regular unipotents. The construction is compatible with the restriction maps and localization techniques already used in GKM theory, and the listed examples demonstrate that the regularity hypothesis is satisfied in several standard settings.

minor comments (3)
  1. [§2] §2: the precise functorial construction of the regular fixed point scheme (its ideal sheaf or its embedding into a product of projective spaces) should be stated before the main theorem is invoked.
  2. [Theorem 5.3] The proof of the isomorphism in the GKM case (Theorem 5.3) relies on a comparison of restriction maps; a short diagram or explicit description of the two maps being identified would improve readability.
  3. [§4] Notation for the coordinate ring of the scheme (e.g., whether it is written as ℂ[Z] or O(Z)) is not uniform across the examples in §4.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a theorem that, under the given regularity hypothesis on the G-action (regular unipotents have finitely many fixed points), the G-equivariant cohomology ring of X is isomorphic to the coordinate ring of an explicitly constructed regular fixed point scheme. This is a direct identification via restriction and localization maps, verified on concrete examples (partial flags, Schubert varieties, Bott-Samelson, GKM spaces). No equation or construction reduces by definition to its own output, no parameter is fitted and then relabeled as a prediction, and no load-bearing step rests on a self-citation whose content is itself unverified. The derivation is therefore self-contained as a standard algebraic-geometry argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract, the result introduces the regular fixed point scheme as the central new object whose coordinate ring matches the cohomology ring; no free parameters are mentioned. The claim rests on standard background results in algebraic geometry and equivariant cohomology.

axioms (1)
  • standard math Standard properties of equivariant cohomology rings for smooth projective varieties and schemes over complex numbers
    The isomorphism is stated to hold in this setting, relying on established definitions and theorems in the field.
invented entities (1)
  • regular fixed point scheme no independent evidence
    purpose: Object whose coordinate ring is defined to be isomorphic to the G-equivariant cohomology ring under regular actions
    Defined in the paper to realize the spectrum of the cohomology as a scheme; no independent evidence outside the isomorphism is provided in the abstract.

pith-pipeline@v0.9.0 · 5614 in / 1204 out tokens · 30917 ms · 2026-05-24T10:25:43.216231+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    Thesis, University of British Columbia, 1977

    Ersan Akyildiz, Geometry of holomorphic vector fields and applications of Gm-actions to linear algebraic groups, Ph.D. Thesis, University of British Columbia, 1977

    work page 1977

  2. [2]

    Ersan Akyildiz, A vector field with one zero on G{P , Proc. Amer. Math. Soc. 67 (1977), 32–34

    work page 1977

  3. [3]

    Akyildiz and J

    E. Akyildiz and J. B. Carrell, Cohomology of projective varieties with regular SL2 action, Manuscripta Math. 58 (1987), 473–486

    work page 1987

  4. [4]

    Akyildiz and J

    E. Akyildiz and J. B. Carrell, A generalization of the Kostant-Macdonald identity , Proc. Natl. Acad. Sci. USA 86 (1989), 3934–3937

    work page 1989

  5. [5]

    Akyildiz, J

    E. Akyildiz, J. B. Carrell and D. I. Lieberman, Zeros of holomorphic vector fields on singular spaces and intersection rings of Schubert varieties , Compos. Math. 57 (1986), 237–248

    work page 1986

  6. [6]

    Akyildiz, J

    E. Akyildiz, J. B. Carrell, D. I. Lieberman and A. J. Sommese, On the graded rings associated to holomorphic vector fields with exactly one zero . In Singularities, Part 1 , Proc. Sympos. Pure Math. 40, 55–56. Amer. Math. Soc., Providence, R.I., 1981

    work page 1981

  7. [7]

    Baum and R

    P. Baum and R. Bott, On the zeros of meromorphic vector-fields . In Essays on Topology and Related Topics (M´ emoires d´ edi´ es ` a Georges de Rham), 29–47. Springer, New York, 1970

    work page 1970

  8. [8]

    D. J. Benson, Polynomial Invariants of Finite Groups , volume 190 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1993

    work page 1993

  9. [9]

    Bia lynicki-Birula,Some theorems on actions of algebraic groups , Ann

    A. Bia lynicki-Birula,Some theorems on actions of algebraic groups , Ann. Math. 98 (1973), 480–497

    work page 1973

  10. [10]

    Bia lynicki-Birula,On fixed points of torus actions on projective varieties , Bull

    A. Bia lynicki-Birula,On fixed points of torus actions on projective varieties , Bull. Acad. Polon. Sci. S´ er. Sci. Math. Astronom. Phys.22 (1974), 1097–1101

    work page 1974

  11. [11]

    Second Enlarged Edition, volume 126 of Graduate Texts in Mathematics

    Armand Borel, Linear Algebraic Groups. Second Enlarged Edition, volume 126 of Graduate Texts in Mathematics. Springer-Verlag, 1991

    work page 1991

  12. [12]

    Michigan Math

    Raoul Bott, Vector fields and characteristic numbers . Michigan Math. J. 14 (1967), 231–244

    work page 1967

  13. [13]

    Raoul Bott and Hans Samelson, Applications of the theory of Morse to symmetric spaces , Amer. J. Math. 80 (1958), 964–1029

    work page 1958

  14. [14]

    Springer, 1989

    Nicolas Bourbaki, Lie Groups and Lie Algebras: Chapters 1–3 . Springer, 1989

    work page 1989

  15. [15]

    In Pa- rameter Spaces, volume 36 of Banach Center Publications , 25–44

    Michel Brion, Piecewise polynomial functions, convex polytopes and enumerative geometry . In Pa- rameter Spaces, volume 36 of Banach Center Publications , 25–44. Institute of Mathematics, Polish Academy of Sciences, 1996

    work page 1996

  16. [16]

    Carrell, The equivariant cohomology ring of regular varieties , Michigan Math

    Michel Brion and James B. Carrell, The equivariant cohomology ring of regular varieties , Michigan Math. J. 52 (2004), 189–203

    work page 2004

  17. [17]

    J. B. Carrell, Torus actions and cohomology. In Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action , volume 131 of Encyclopaedia of Mathematical Sciences, 83–158. Springer, Berlin, 2002

    work page 2002

  18. [18]

    Carrell, Deformation of the nilpotent zero scheme and the intersection ring of invariant subvarieties, J

    James B. Carrell, Deformation of the nilpotent zero scheme and the intersection ring of invariant subvarieties, J. reine. angew. Math. 460 (1995), 37–54. 57

    work page 1995

  19. [19]

    J. B. Carrell and Kiumars Kaveh, On the equivariant cohomology of subvarieties of a B-regular variety, Transform. Groups 13 (2008), 495–505

    work page 2008

  20. [20]

    J. B. Carrell and D. I. Liebermann, Vector fields and Chern numbers , Math. Ann. 225 (1977), 263–267

    work page 1977

  21. [21]

    Tome III: Th´ eor` emes G´ en´ eraux sur les Alg` ebres de Lie

    Claude Chevalley, Th´ eorie des Groupes de Lie. Tome III: Th´ eor` emes G´ en´ eraux sur les Alg` ebres de Lie. Volume IV of Publications de l’Institut Math´ ematique de l’Universit´ e de Nancago. Librairie Scientifique Hermann et Cie, 1955

    work page 1955

  22. [22]

    Birkh¨ auser, 1997

    Neil Chriss and Victor Ginzburg, Representation Theory and Complex Geometry . Birkh¨ auser, 1997

    work page 1997

  23. [23]

    Cohen and Jan Draisma, From Lie algebras of vector fields to algebraic group actions

    Arjeh M. Cohen and Jan Draisma, From Lie algebras of vector fields to algebraic group actions . Transform. Groups. 8 (2003), 51–68

    work page 2003

  24. [24]

    Michel Demazure, D´ esingularisation des vari´ et´ es de Schubert g´ en´ eralis´ ees, Ann. Sci. ´Ecole Norm. Sup. 4 (1974), 53–88

    work page 1974

  25. [25]

    D. ˇZ. Djokovi´ c and J.Malzan,Orbits of Nilpotent Matrices , Linear Algebra Appl. 32 (1980), 157–158

    work page 1980

  26. [26]

    Volume 296 of London Mathematical Society Lecture Note Series

    Igor Dolgachev, Lectures on Invariant Theory. Volume 296 of London Mathematical Society Lecture Note Series. Cambridge University Press, 2010

    work page 2010

  27. [27]

    Volume 150 of Graduate Texts in Mathematics

    David Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry . Volume 150 of Graduate Texts in Mathematics . Springer-Verlag, 1995

    work page 1995

  28. [28]

    Springer, 1998

    William Fulton, Intersection Theory, Second Edition. Springer, 1998

    work page 1998

  29. [29]

    Wee Liang Gan and Victor Ginzburg, Quantization of Slodowy slices , Int. Math. Res. Not. IMRN 5 (2002), 243–255

    work page 2002

  30. [30]

    Victor Ginzburg, Principal nilpotent pairs in a semisimple Lie algebra 1 , Invent. math. 140 (2000), 511–561

    work page 2000

  31. [31]

    Mark Goresky and Robert MacPherson, On the spectrum of the equivariant cohomology ring , Canad. J. Math. 62 (2010), no. 2, 262–283

    work page 2010

  32. [32]

    Mark Goresky, Robert Kottwitz and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. math. 131 (1998), 25–83

    work page 1998

  33. [33]

    Vieweg+Teubner, 2010

    Ulrich G¨ ortz and Torsten Wedhorn,Algebraic Geometry I: Schemes With Examples and Exercises . Vieweg+Teubner, 2010

    work page 2010

  34. [34]

    Alexander Grothendieck and Jean Dieudonn´ e,´El´ ements de g´ eom´ etrie alg´ ebraique: III.´Etude coho- mologique des faisceaux coh´ erents, Premi` ere partie, Publ. Math. IH ´ES 11 (1961), 5–167

    work page 1961

  35. [35]

    Alexander Grothendieck and Jean Dieudonn´ e,´El´ ements de g´ eom´ etrie alg´ ebraique: IV.´Etude locale des sch´ emas et des morphismes de sch´ emas, Troisi` eme partie, Publ. Math. IH ´ES 28 (1966), 5–255

    work page 1966

  36. [36]

    H. C. Hansen, On cycles in flag manifolds , Math. Scand. 33 (1974), 269–274

    work page 1974

  37. [37]

    Volume 133 of Graduate Texts in Mathematics

    Joe Harris, Algebraic Geometry: A First Course . Volume 133 of Graduate Texts in Mathematics . Springer-Verlag, 1992. 58

    work page 1992

  38. [38]

    Landweber and Reyer Sjamaar, Divided differences and the Weyl character formula in equivariant K-theory , Math

    Megumi Harada, Gregory D. Landweber and Reyer Sjamaar, Divided differences and the Weyl character formula in equivariant K-theory , Math. Res. Lett. 17 (2010), 507–527

    work page 2010

  39. [39]

    Corrected eighth printing, volume 52 of Graduate Texts in Mathematics

    Robin Hartshorne, Algebraic Geometry. Corrected eighth printing, volume 52 of Graduate Texts in Mathematics. Springer-Verlag, 1997

    work page 1997

  40. [40]

    Tam´ as Hausel,Big algebras, arXiv:2311.02711

    work page arXiv

  41. [41]

    Tam´ as Hausel,The Hitchin map as spectra of equivariant cohomology and Kirillov algebras (in preparation)

    work page

  42. [42]

    Tatsuyuki Hikita, An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane. Int. Math. Res. Not. IMRN 2017, no. 8, 2538–2561

    work page 2017

  43. [43]

    Horrocks, Fixed point schemes of additive group actions , Topology 8 (1969), 233–242

    G. Horrocks, Fixed point schemes of additive group actions , Topology 8 (1969), 233–242

    work page 1969

  44. [44]

    Volume 85 of Ergebnisse der Mathematik und ihrer Grenzgebiete

    Wu Yi Hsiang, Cohomology Theory of Topological Transformation Groups. Volume 85 of Ergebnisse der Mathematik und ihrer Grenzgebiete . Springer-Verlag, 1975

    work page 1975

  45. [45]

    Humphreys, Introduction to Lie Algebras and Representation Theory

    James E. Humphreys, Introduction to Lie Algebras and Representation Theory . Volume 9 of Graduate Texts in Mathematics. Springer-Verlag, 1972

    work page 1972

  46. [46]

    Humphreys, Linear Algebraic Groups

    James E. Humphreys, Linear Algebraic Groups. Corrected fourth printing, volume 21 of Graduate Texts in Mathematics. Springer-Verlag, 1995

    work page 1995

  47. [47]

    Dover Publications, 1979

    Nathan Jacobson, Lie Algebras. Dover Publications, 1979

    work page 1979

  48. [48]

    Bertram Kostant, Lie Group representations on polynomial rings , Amer. J. Math. 85 (1963), 327–404

    work page 1963

  49. [49]

    Bertram Kostant, On Whittaker vectors and representation theory , Invent. Math. 48 (1978), 101–184

    work page 1978

  50. [50]

    Shrawan Kumar and Claudio Procesi, An algebro-geometric realization of equivariant cohomology of some Springer fibers, J. Algebra. 368 (2012), 70–74

    work page 2012

  51. [51]

    Jakub L¨ owit,private communication

    work page

  52. [52]

    Translated by M

    Hideyuki Matsumura, Commutative Ring Theory. Translated by M. Reid. Ninth printing, volume 8 of Cambridge studies in advanced mathematics . Cambridge University Press, 2006

    work page 2006

  53. [53]

    Alexander Merkurjev, Comparison of the equivariant and the standard K-theory of algebraic varieties. , St. Petersburg Math. J. 9 (1998), 815–850

    work page 1998

  54. [54]

    J. S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field . Volume 170 of Cambridge studies in advanced mathematics . Cambridge University Press, 2017

    work page 2017

  55. [55]

    G. D. Mostow, Fully Reducible Subgroups of Algebraic Groups , Amer. J. Math. 78 (1956), 200–221

    work page 1956

  56. [56]

    Vanchinathan, Small Resolutions of Schubert Varieties in Symplectic and Orthogonal Grassmannians, Publ

    Parameswaran Sankaran and P. Vanchinathan, Small Resolutions of Schubert Varieties in Symplectic and Orthogonal Grassmannians, Publ. RIMS, Kyoto Univ. 30 (1994), 443–458

    work page 1994

  57. [57]

    T. A. Springer, Linear Algebraic Groups. Second Edition, Springer, 1998

    work page 1998

  58. [58]

    Robert Steinberg, Torsion in Reductive Groups. Adv. Math. 15 (1975), 63–92. 59

    work page 1975

  59. [59]

    R. W. Thomason, XX. Algebraic K-Theory of Group Scheme Actions . In Algebraic Topology and Algebraic K-Theory, volume 113 of Annals of Mathematics Studies , 539–563. Princeton University Press, 2016

    work page 2016

  60. [60]

    Matthieu Willems, Cohomologie et K-th´ eorie ´ equivariantes des vari´ et´ es de Bott–Samelson et des vari´ et´ es de drapeaux, Bull. Soc. Math. France 132 (2004), 569–589. 60

    work page 2004