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arxiv: 2604.23997 · v1 · submitted 2026-04-27 · 🧮 math.AG

The Birational Invariance of Fundamental Group Schemes

Lingguang Li , Hao Wang This is my paper

Pith reviewed 2026-05-08 02:26 UTC · model grok-4.3

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The pith

Various fundamental group schemes are birationally invariant for smooth projective varieties over perfect fields.

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

In algebraic geometry, varieties are shapes defined by polynomial equations. The fundamental group scheme is an algebraic version of the topological fundamental group, built from categories of vector bundles or sheaves via Tannakian duality, which turns the category into a group scheme. The authors give general criteria under which a birational morphism between proper schemes induces an isomorphism on these group schemes. They apply the criteria to show that many specific versions, including etale, unipotent, and others, are the same for birationally equivalent smooth projective varieties over perfect fields.

Core claim

For a birational map X ⇢ Y between smooth projective varieties over a perfect field k, there exists a natural isomorphism π^*(X,x) ≅ π^*(Y,y) for any * ∈ {S,N,EN,F,EF,Loc,ELoc,ét, Eét,uni}. In particular, the induced homomorphism π^str(X,x) → π^str(Y,y) is an isomorphism for any birational morphism X → Y.

Load-bearing premise

Y is normal, the schemes are integral connected and proper over k, and the Tannakian categories C_X and C_Y satisfy the general criteria making the natural homomorphism an isomorphism; for the main application the varieties must be smooth projective over a perfect field.

read the original abstract

Let $k$ be a field, $f \colon X \to Y$ a birational morphism of integral connected schemes proper over $k$ with $Y$ normal, $x \in X(k)$ lying over $y \in Y(k)$. For Tannakian categories $\cC_X \subset \Vect(X)$ and $\cC_Y \subset \Vect(Y)$, denote by $\pi(\cC_X,x)$ and $\pi(\cC_Y,y)$ the corresponding Tannaka group schemes. We establish general Tannakian criteria for the natural homomorphism $\pi(\cC_X,x)\to \pi(\cC_Y,y)$ to be an isomorphism. As applications, for a birational map $X \dashrightarrow Y$ between smooth projective varieties over a perfect field $k$, we prove that there exists a natural isomorphism $\pi^{*}(X,x)\cong \pi^{*}(Y,y)$ for any $* \in \{S,N,EN,F,EF,Loc,ELoc,\acute{e}t, E\acute{e}t,uni\}$. In particular, we prove that the induced homomorphism $\pi^{str}(X,x)\to \pi^{str}(Y,y)$ is an isomorphism for any birational morphism $ X \rightarrow Y$.

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard Tannakian duality and scheme-theoretic properties with no new free parameters or invented entities visible in the abstract.

axioms (2)
  • standard math Tannakian duality applies to the relevant categories of vector bundles or sheaves on the schemes
    This is the foundational framework used to define the group schemes π(C,x).
  • domain assumption The natural homomorphism induced by the birational morphism between the Tannakian categories is an isomorphism under the stated conditions on normality and properness
    This is the key premise invoked to obtain the isomorphism criteria.

pith-pipeline@v0.9.0 · 5522 in / 1330 out tokens · 72100 ms · 2026-05-08T02:26:25.131862+00:00 · methodology

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Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Adroja and S

    P. Adroja and S. Amrutiya,On an extension of Nori and local fundamental group schemes, Comm. Algebra53(2025), no. 10, 4241–4255

    work page 2025

  2. [2]

    Amrutiya, I

    S. Amrutiya, I. Biswas,On the F-fundamental group scheme, Bull. Sci. Math.134(2010), no. 5, 461–474

    work page 2010

  3. [3]

    Amrutiya,A note on certain Tannakian group schemes, Archivum Mathematicum56(2020), no

    S. Amrutiya,A note on certain Tannakian group schemes, Archivum Mathematicum56(2020), no. 1, 21–29

    work page 2020

  4. [4]

    Chatzistamatiou and K

    A. Chatzistamatiou and K. R¨ ulling,Higher direct images of the structure sheaf in positive characteristic, Algebra Number Theory.5(2011), no. 6, 693–775

    work page 2011

  5. [5]

    Cheng, C

    R. Cheng, C. Lian and T. Murayama, Projectivity of the moduli of curves, inStacks Project Expository Collection (SPEC), 1–43, London Math. Soc. Lecture Note Ser., 480, Cambridge Univ. Press, Cambridge (2022)

    work page 2022

  6. [6]

    J. P. dos Santos,Fundamental group schemes for stratified sheaves, J. Algebra317(2007), no. 2, 691–713

    work page 2007

  7. [7]

    Grothendieck,Revˆ etements ´ etales et groupe fondamental, S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois-Marie, (SGA 1), 1960/61, Lecture Notes in Mathematics, Vol

    A. Grothendieck,Revˆ etements ´ etales et groupe fondamental, S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois-Marie, (SGA 1), 1960/61, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin (1971)

    work page 1960

  8. [8]

    Hogadi and V

    A. Hogadi and V. B. Mehta, Birational invariance of theS-fundamental group scheme, Pure Appl. Math. Q.7(2011), no. 4, Special Issue: In memory of Eckart Viehweg, 1361–1369

    work page 2011

  9. [9]

    Langer,On the S-fundamental group scheme, Ann

    A. Langer,On the S-fundamental group scheme, Ann. Inst. Fourier (Grenoble)61(2011), no. 5, 2077–2119 (2012)

    work page 2011

  10. [10]

    L. Li, N. Tian,The base change of fundamental group schemes,https://doi.org/10.48550/arXiv.2602.11110

    work page doi:10.48550/arxiv.2602.11110

  11. [11]

    L. Li, N. Tian,The K¨ unneth formula of fundamental group schemes,https://doi.org/10.48550/arXiv.2602.14207. 8 LINGGUANG LI AND HAO WANG

    work page doi:10.48550/arxiv.2602.14207

  12. [12]

    V. B. Mehta and S. Subramanian,Some remarks on the local fundamental group scheme, Proc. Indian Acad. Sci. Math. Sci.118(2008), no. 2, 207–211

    work page 2008

  13. [13]

    M. V. Nori,The fundamental group-scheme, Proc. Indian Acad. Sci. Math. Sci.91(1982), no. 2, 73–122

    work page 1982

  14. [14]

    Otabe,An extension of Nori fundamental group, Comm

    S. Otabe,An extension of Nori fundamental group, Comm. Algebra45(2017), no. 8, 3422–3448

    work page 2017

  15. [15]

    The Stacks project authors,The Stacks project,https://stacks.math.columbia.edu, 2026

    work page 2026

  16. [16]

    Sun and L

    X. Sun and L. Zhang,The ´ etale fundamental group andF-divided sheaves in characteristicp >0,https://doi.org/ 10.48550/arXiv.2509.24360

    work page doi:10.48550/arxiv.2509.24360

  17. [17]

    Yekutieli,Derived Categories, Cambridge Studies in Advanced Mathematics, Vol

    A. Yekutieli,Derived Categories, Cambridge Studies in Advanced Mathematics, Vol. 183, Cambridge University Press (2019). School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Shanghai 200092, CHINA Email address:LiLg@tongji.edu.cn School of Mathematical Sciences, Key Laborator...

    work page 2019