pith. sign in

arxiv: 2507.13452 · v2 · pith:GZX4KTHTnew · submitted 2025-07-17 · 🧮 math.AG

Generic vanishing on homogeneous spaces in arbitrary characteristic

Ankit Rai , K. V. Shuddhodan This is my paper

Pith reviewed 2026-05-21 23:36 UTC · model grok-4.3

classification 🧮 math.AG
keywords homogeneous spacesEuler characteristicgeneric vanishingalgebraic groupsarbitrary characteristicintersection theorytrace functions
0
0 comments X

The pith

For generic group elements the signed Euler characteristic of intersections of smooth affine subvarieties on proper homogeneous spaces is non-negative in arbitrary characteristic.

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

The paper shows that on a proper homogeneous space X for a connected algebraic group G over an algebraically closed field, if W and Z are smooth locally closed affine subvarieties, then the signed Euler characteristic of their generic intersection is non-negative. Specifically, (-1) to the power of dim X minus dim W plus dim Z times chi of gW intersect Z is at least zero for generic g in G. This result holds in any characteristic and extends a prior theorem from characteristic zero. It is of interest because it gives a uniform way to control intersection Euler characteristics using group translations, and provides additional identities and estimates when the field is finite.

Core claim

Let X be a proper homogeneous space for a connected algebraic group G over an algebraically closed field. For locally closed smooth affine subvarieties W and Z in X, the inequality (-1)^{dim X - dim W + dim Z} χ(gW ∩ Z) ≥ 0 holds for generic g in G. The proof works in arbitrary characteristic, and when the field is finite it yields a trace-function identity on a dense open subset of G and a Lang-Weil estimate for the non-generic locus.

What carries the argument

The signed Euler characteristic of the intersection gW ∩ Z under generic group translation, controlled via smoothness and affinity assumptions to ensure the sign is determined by dimension.

If this is right

  • The inequality applies equally in positive characteristic as in zero.
  • Over finite fields a trace function identity holds on a dense open in G.
  • A Lang-Weil estimate bounds the non-generic locus in G.

Where Pith is reading between the lines

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

  • This suggests that Euler characteristic positivity under group action may be independent of characteristic in homogeneous settings.
  • Similar results could apply to other proper varieties with group actions if smoothness is relaxed via resolutions.

Load-bearing premise

W and Z are smooth and affine while X is proper, which lets the proof manage intersection dimensions and use characteristic-independent methods.

What would settle it

A concrete counterexample consisting of a proper homogeneous space, a connected algebraic group, and two smooth affine subvarieties where the signed Euler characteristic of the intersection is negative for a generic group element.

read the original abstract

Let $X$ be a proper homogeneous space for a connected algebraic group $G$ over an algebraically closed field. For locally closed smooth affine subvarieties $W,Z\subset X$, we show that \[ (-1)^{\dim X-\dim W+\dim Z}\chi(gW\cap Z)\geq 0 \] for generic $g\in G$. This extends the characteristic-zero theorem of Sch\"urmann--Simpson--Wang. Over finite fields, our methods give a trace-function identity on a dense open subset of $G$ and a Lang--Weil estimate for the non-generic locus.

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 proves a generic vanishing result for Euler characteristics on proper homogeneous spaces: Let X be a proper homogeneous space for a connected algebraic group G over an algebraically closed field. For locally closed smooth affine subvarieties W, Z ⊂ X, the inequality (-1)^{dim X - dim W + dim Z} χ(gW ∩ Z) ≥ 0 holds for generic g ∈ G. This extends the characteristic-zero theorem of Schürmann-Simpson-Wang to arbitrary characteristic. Over finite fields the methods additionally yield a trace-function identity on a dense open subset of G together with a Lang-Weil estimate controlling the non-generic locus.

Significance. If the central inequality holds, the result supplies a useful extension of generic vanishing theorems into positive characteristic, with direct consequences for intersection theory on homogeneous spaces. The finite-field trace identity and Lang-Weil bound constitute concrete arithmetic output that strengthens the contribution. The manuscript is credited for developing new techniques adapted to positive characteristic rather than merely reducing to the characteristic-zero case.

minor comments (3)
  1. The abstract states the main inequality clearly but does not indicate the precise location in the text where the positive-characteristic vanishing argument is completed; a forward reference would help readers.
  2. In the finite-field section, the statement of the trace-function identity should explicitly record the dense open subset of G on which it holds, rather than leaving the locus implicit.
  3. A short comparison paragraph situating the new positive-characteristic methods against the original Schürmann-Simpson-Wang argument would clarify the novelty for readers familiar with the characteristic-zero case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the recognition of the extension to arbitrary characteristic, and the recommendation for minor revision. We are pleased that the new techniques and the finite-field arithmetic consequences are viewed as strengthening the contribution.

Circularity Check

0 steps flagged

No significant circularity; extends external theorem with independent methods

full rationale

The paper states a signed Euler characteristic inequality for generic intersections of smooth affine locally closed subvarieties on proper homogeneous spaces, explicitly extending the characteristic-zero result of Schürmann-Simpson-Wang. The abstract and available description indicate use of new methods for positive characteristic, including trace-function identities and Lang-Weil estimates over finite fields. No load-bearing self-citations, self-definitional steps, or fitted parameters renamed as predictions are present in the provided text. The central claim relies on external prior work and smoothness/affineness assumptions without reducing to its own inputs by construction. This is the expected non-finding for a paper whose derivation chain builds on independent external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the prior characteristic-zero theorem plus new arguments for positive characteristic; no free parameters or invented entities are visible from the abstract.

axioms (2)
  • domain assumption X is a proper homogeneous space for a connected algebraic group G over an algebraically closed field.
    Stated in the abstract as the setting for the result.
  • domain assumption W and Z are locally closed smooth affine subvarieties.
    Required for the intersection χ to satisfy the signed inequality.

pith-pipeline@v0.9.0 · 5621 in / 1269 out tokens · 33570 ms · 2026-05-21T23:36:13.449347+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

20 extracted references · 20 canonical work pages

  1. [1]

    Beilinson, J

    A. Beilinson, J. Bernstein, P. Deligne, and O. Gabber. Faisceaux pervers. Soci \'e t \'e math \'e matique de France , 100, 2018

    work page 2018

  2. [2]

    A. A. Beilinson. On the derived category of perverse sheaves. In K -theory, arithmetic and geometry ( M oscow, 1984--1986) , volume 1289 of Lecture Notes in Math. , pages 27--41. Springer, Berlin, 1987

    work page 1984

  3. [3]

    Transformations canoniques, dualit\' e projective, th\' e orie de L efschetz, transformations de F ourier et sommes trigonom\' e triques

    Jean-Luc Brylinski. Transformations canoniques, dualit\' e projective, th\' e orie de L efschetz, transformations de F ourier et sommes trigonom\' e triques. Number 140-141, pages 3--134, 251. 1986. G\' e om\' e trie et analyse microlocales

    work page 1986

  4. [4]

    Vanishing theorems for perverse sheaves on abelian varieties, revisited

    Bhargav Bhatt, Christian Schnell, and Peter Scholze. Vanishing theorems for perverse sheaves on abelian varieties, revisited. Selecta Mathematica , 24(1):63--84, 2018

    work page 2018

  5. [5]

    P. Deligne. Cohomologie \' E tale , volume 569 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1977. S\' e minaire de g\' e om\' e trie alg\' e brique du Bois-Marie SGA 4 1 2

    work page 1977

  6. [6]

    La conjecture de W eil

    Pierre Deligne. La conjecture de W eil. II . Inst. Hautes \' E tudes Sci. Publ. Math. , (52):137--252, 1980

    work page 1980

  7. [7]

    M. A. deCataldo and L. Migliorini. The perverse filtration and the L efschetz hyperplane theorem. Annals of Mathematics , 171:1797--1819, 2010

    work page 2010

  8. [8]

    Forey, J

    Arthur Forey, Javier Fres \'a n, and Emmanuel Kowalski. Arithmetic F ourier transforms over finite fields: generic vanishing, convolution, and equidistribution. arXiv preprint arXiv:2109.11961 , 2021

    work page arXiv 2021

  9. [9]

    Pieri and M urnaghan-- N akayama type rules for C hern classes of S chubert cells

    Neil JY Fan, Peter L Guo, and Rui Xiong. Pieri and M urnaghan-- N akayama type rules for C hern classes of S chubert cells. arXiv preprint arXiv:2211.06802 , 2022

    work page arXiv 2022

  10. [10]

    \'E l \'e ments de g \'e om \'e trie alg \'e brique

    Alexander Grothendieck and Jean Dieudonné. \'E l \'e ments de g \'e om \'e trie alg \'e brique. IV. \'E tude locale des sch \'e mas et des morphismes de sch \'e mas. Quatri \`e me partie , volume 32 of Publications Math \'e matiques de l'IH \'E S . Institut des Hautes \'E tudes Scientifiques, Bures-sur-Yvette, 1967

    work page 1967

  11. [11]

    Deformation theory, generic vanishing theorems, and some conjectures of E nriques, C atanese and B eauville

    Mark Green and Robert Lazarsfeld. Deformation theory, generic vanishing theorems, and some conjectures of E nriques, C atanese and B eauville. Inventiones mathematicae , 90(2):389--407, 1987

    work page 1987

  12. [12]

    Faisceaux pervers -adiques sur un tore

    Ofer Gabber and Fran c ois Loeser. Faisceaux pervers -adiques sur un tore. Duke Mathematical Journal , 83(3):501--606, 1996

    work page 1996

  13. [13]

    Nicholas M. Katz. Affine cohomological transforms, perversity, and monodromy. J. Amer. Math. Soc. , 6(1):149--222, 1993

    work page 1993

  14. [14]

    Conjectural positivity of C hern-- S chwartz-- M ac P herson classes for R ichardson cells

    Shrawan Kumar. Conjectural positivity of C hern-- S chwartz-- M ac P herson classes for R ichardson cells. International Mathematics Research Notices , 2024(2):1154--1165, 2024

    work page 2024

  15. [15]

    Schubert calculus and quiver varieties

    Allen Knutson and Paul Zinn-Justin. Schubert calculus and quiver varieties. Proc. ICM (2022, virtual) , 6:4582--4605, 2022

    work page 2022

  16. [16]

    G. Laumon. Comparaison de caracteristiques d' E uler- P oincare en cohomologie -adique. C. R. Acad. Sci. Paris, Serie I , 292:209--212, 1981

    work page 1981

  17. [17]

    Holonomic D -modules on abelian varieties

    Christian Schnell. Holonomic D -modules on abelian varieties. Publications M ath \'e matiques de l'IH \'E S , 121(1):1--55, 2015

    work page 2015

  18. [18]

    o rg Sch \

    J \"o rg Sch \"u rmann, Connor Simpson, and Botong Wang. A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of S chubert cells. Compositio Mathematica , 161(1):1--12, 2025

    work page 2025

  19. [19]

    Stacks Project

    The Stacks Project Authors . Stacks Project. https://stacks.math.columbia.edu, 2025

    work page 2025

  20. [20]

    Vanishing theorems for constructible sheaves on abelian varieties over finite fields

    Rainer Weissauer. Vanishing theorems for constructible sheaves on abelian varieties over finite fields. Math. Ann. , 365(1-2):559--578, 2016

    work page 2016