pith. sign in

arxiv: 2404.08840 · v3 · submitted 2024-04-12 · 🧮 math.DG

On Nash resolution of (singular) Lie algebroids

Ruben Louis This is my paper

Pith reviewed 2026-05-24 02:24 UTC · model grok-4.3

classification 🧮 math.DG
keywords Lie algebroidsNash blow-upsingular foliationsblow-upsexact sequencesLie algebra bundlessingular subalgebroids
0
0 comments X

The pith

Any Lie algebroid admits a Nash-type blow-up fitting into a short exact sequence of Lie algebroids.

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

The paper shows that for any Lie algebroid A, a Nash-type blow-up Nash(A) can be constructed by blowing up the base variety along the singular foliation determined by A. This blow-up fits into the short exact sequence 0 → K → Nash(A) → D → 0, where K is a Lie algebra bundle and the anchor of D is injective on an open dense subset. A reader would care if this provides a standard way to desingularize Lie algebroids while keeping their algebraic structure intact. The result is illustrated with examples and extended to singular subalgebroids.

Core claim

Any Lie algebroid A admits a Nash-type blow-up Nash(A) that sits in a short exact sequence of Lie algebroids 0→K→Nash(A)→D→0 with K a Lie algebra bundle and D a Lie algebroid whose anchor map is injective on an open dense subset. The base variety is a blowup determined by the singular foliation of A. The construction extends to singular subalgebroids.

What carries the argument

The Nash-type blow-up Nash(A) of a Lie algebroid A, obtained via blowup of the base along its singular foliation, which resolves the structure into the exact sequence with Lie algebra bundle kernel.

If this is right

  • The anchor map of the quotient D is injective on an open dense subset of the blown-up base.
  • K is a Lie algebra bundle in the exact sequence.
  • The construction applies to singular subalgebroids.
  • Concrete examples of the resolution are given for specific Lie algebroids.

Where Pith is reading between the lines

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

  • The resolution may allow transferring properties from regular Lie algebroids to singular ones via the blow-up.
  • Similar techniques could apply to other singular geometric objects like singular foliations or Poisson structures.
  • Further iterations of the blow-up might be needed if D retains some singularities.

Load-bearing premise

The blowup of the base variety determined by the singular foliation carries a compatible Lie algebroid structure satisfying the exact sequence.

What would settle it

An explicit Lie algebroid where the proposed Nash blow-up either does not exist or fails to produce a quotient with injective anchor on a dense open set.

read the original abstract

Any Lie algebroid $A$ admits a Nash-type blow-up $\mathrm{Nash}(A)$ that sits in a nice short exact sequence of Lie algebroids $0\rightarrow K\rightarrow \mathrm{Nash}(A)\rightarrow \mathcal{D}\rightarrow 0$ with $K$ a Lie algebra bundle and $\mathcal{D}$ a Lie algebroid whose anchor map is injective on an open dense subset. The base variety is a blowup determined by the singular foliation of $A$. We provide concrete examples. Moreover, we extend the construction following Mohsen's to singular subalgebroids in the sense of Androulidakis-Zambon.

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

2 major / 1 minor

Summary. The manuscript claims that any Lie algebroid A admits a Nash-type blow-up Nash(A) of its base along the singular foliation, fitting into the short exact sequence 0 → K → Nash(A) → D → 0 of Lie algebroids (K a Lie algebra bundle, D with injective anchor on a dense open set). Concrete examples are supplied and the construction is extended to singular subalgebroids in the sense of Androulidakis-Zambon.

Significance. If the existence and exact sequence are established, the result supplies a canonical resolution procedure for singular Lie algebroids that could be useful in the study of singular foliations and related geometric structures.

major comments (2)
  1. [Abstract, §1] Abstract and §1: the existence of Nash(A) and the short exact sequence are asserted, but the construction of the blow-up and the verification that the lifted anchor and bracket satisfy the Lie algebroid axioms (so that the sequence is exact in the category of Lie algebroids) are not outlined; this is load-bearing for the central claim.
  2. [Examples] Examples section: the concrete examples are stated to illustrate the sequence, yet no explicit check is supplied that the anchor of D is injective on a dense open set or that K is indeed a Lie algebra bundle; this verification is required to support the general statement.
minor comments (1)
  1. Notation for the blow-up center and the foliation ideal should be introduced uniformly before the main construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments. We address each major comment below and will make the indicated revisions to improve clarity.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: the existence of Nash(A) and the short exact sequence are asserted, but the construction of the blow-up and the verification that the lifted anchor and bracket satisfy the Lie algebroid axioms (so that the sequence is exact in the category of Lie algebroids) are not outlined; this is load-bearing for the central claim.

    Authors: We agree that the abstract and introduction would benefit from a concise outline of the Nash blow-up construction together with the main steps verifying that the lifted anchor and bracket satisfy the Lie algebroid axioms and that the sequence is exact in the category of Lie algebroids. While the full details appear in Section 3, we will insert a short summary of the construction and the key verifications into the revised §1 and abstract. revision: yes

  2. Referee: [Examples] Examples section: the concrete examples are stated to illustrate the sequence, yet no explicit check is supplied that the anchor of D is injective on a dense open set or that K is indeed a Lie algebra bundle; this verification is required to support the general statement.

    Authors: We acknowledge that the examples would be strengthened by explicit verification that the anchor of D is injective on a dense open set and that K is a Lie algebra bundle in each case. We will add these direct checks to the revised examples section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is an explicit geometric lift

full rationale

The paper presents an existence construction: for any Lie algebroid A, form the blow-up of the base along the singular foliation of A, then lift the Lie algebroid structure to obtain Nash(A) fitting into the short exact sequence 0→K→Nash(A)→D→0. The abstract and claim rest on standard blow-up geometry and anchor/bracket lifting, with an extension to singular subalgebroids following Mohsen (external citation, not self-citation). No equation reduces a 'prediction' to a fitted parameter by construction, no load-bearing self-citation chain, and no ansatz smuggled via prior work by the same author. The derivation is self-contained against external geometric notions and does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of Lie algebroids and the existence of a foliation-determined blow-up; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)
  • standard math Lie algebroids are vector bundles with bracket and anchor satisfying the usual Leibniz and Jacobi identities.
    Background definition invoked throughout the abstract.
  • domain assumption The singular foliation of A determines a blow-up of the base on which the Nash algebroid can be defined.
    Explicitly stated as the base of the construction in the abstract.

pith-pipeline@v0.9.0 · 5627 in / 1300 out tokens · 24058 ms · 2026-05-24T02:24:53.802247+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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A series of Nash resolutions of a singular foliation

    math.DG 2023-01 unverdicted novelty 5.0

    Constructs a series of Nash blowups of singular foliations that turn any such foliation into a Debord foliation after one step, recovering prior cases for i=0 and i=1.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Aldrovandi, U

    E. Aldrovandi, U. Bruzzo, and V. Rubtsov. Lie algebroid cohomology and L ie algebroid extensions. Journal of Algebra , 505:456--481, 2018

    work page 2018

  2. [2]

    The holonomy groupoid of a singular foliation

    Iakovos Androulidakis and Georges Skandalis. The holonomy groupoid of a singular foliation. J. Reine Angew. Math. , 626:1--37, 2009

    work page 2009

  3. [3]

    Splitting theorems for Poisson and related structures

    Henrique Bursztyn , Hudson Lima , and Eckhard Meinrenken . Splitting theorems for Poisson and related structures. J. Reine Angew. Math. , 754:281--312, 2019

    work page 2019

  4. [4]

    Distributions involutives singuli\`eres

    Dominique Cerveau. Distributions involutives singuli\`eres. Ann. Inst. Fourier (Grenoble) , 29(3):xii, 261--294, 1979. http://archive.numdam.org/article/AIF_1979__29_3_261_0.pdf

    work page 1979

  5. [5]

    Integrability of Lie brackets

    Marius Crainic and Rui Loja Fernandes. Integrability of Lie brackets. Ann. Math. (2) , 157(2):575--620, 2003

    work page 2003

  6. [6]

    Lectures on poisson geometry

    Marius Crainic, Rui Loja Fernandes, and Ioan Mărcuţ. Lectures on poisson geometry. Graduate Studies in Mathematics , 2021

    work page 2021

  7. [7]

    Local integration of L ie algebroids

    Claire Debord. Local integration of L ie algebroids. Banach Center Publications , 54:21--33, 2000

    work page 2000

  8. [8]

    Holonomy groupoids of singular foliations

    Claire Debord. Holonomy groupoids of singular foliations. J. Differential Geom. , 58(3):467--500, 2001

    work page 2001

  9. [9]

    Symplectic and poisson geometry on b-manifolds

    Victor Guillemin, Eva Miranda, and Ana Rita Pires. Symplectic and poisson geometry on b-manifolds. Advances in Mathematics , 264:864--896, 2014

    work page 2014

  10. [10]

    On the inner automorphisms of a singular foliation

    Alfonso Garmendia and Ori Yudilevich. On the inner automorphisms of a singular foliation. Mathematische Zeitschrift , 2018

    work page 2018

  11. [11]

    J. Harris. Algebraic Geometry: A First Course . Graduate Texts in Mathematics. Springer, 1992

    work page 1992

  12. [12]

    R. Hermann. The differential geometry of foliations. II . J. Math. Mech. , 11:303--315, 1962

    work page 1962

  13. [13]

    Higher homotopies and M aurer- C artan algebras: Q uasi- L ie- R inehart, G erstenhaber, and B atalin- V ilkovisky algebras

    Johannes Huebschmann. Higher homotopies and M aurer- C artan algebras: Q uasi- L ie- R inehart, G erstenhaber, and B atalin- V ilkovisky algebras. Progress in Mathematics , 232, 2003

    work page 2003

  14. [14]

    Complex geometry: An introduction

    Daniel Huybrechts. Complex geometry: An introduction. 2004. https://api.semanticscholar.org/CorpusID:118020813

    work page 2004

  15. [15]

    Differential operators and actions of lie algebroids

    Yvette Kosmann-Schwarzbach and Kirill CH Mackenzie. Differential operators and actions of lie algebroids. Contemporary Mathematics , 315:213--234, 2002

    work page 2002

  16. [16]

    [LGLR24] Camille Laurent-Gengoux, Ruben Louis, and Leonid Ryvkin

    Camille Laurent-Gengoux and Ruben Louis. Lie- R inehart algebras acyclic L ie -algebroids. Journal of algebra , 594:1--53, 2022. https://doi.org/10.1016/j.jalgebra.2021.11.023

    work page doi:10.1016/j.jalgebra.2021.11.023 2022

  17. [17]

    Canonical Geometric and Algebraic Structures Hidden Behind a Singular Foliation , pages 263--390

    Camille Laurent-Gengoux, Ruben Louis, and Leonid Ryvkin. Canonical Geometric and Algebraic Structures Hidden Behind a Singular Foliation , pages 263--390. Springer Nature Switzerland, Cham, 2025

    work page 2025

  18. [18]

    What Is a Singular Foliation? , pages 123--261

    Camille Laurent-Gengoux, Ruben Louis, and Leonid Ryvkin. What Is a Singular Foliation? , pages 123--261. Springer Nature Switzerland, Cham, 2025

    work page 2025

  19. [19]

    The universal L ie -algebroid of a singular foliation

    Camille Laurent-Gengoux, Sylvain Lavau, and Thomas Strobl. The universal L ie -algebroid of a singular foliation. Doc. Math. , 25:1571--1652, 2020

    work page 2020

  20. [20]

    Poisson structures

    Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke. Poisson structures . Grundlehren der mathematischen Wissenschaften A series of comprehensive studies in mathematics. Springer, Heidelberg New York London, C 2013. https://link.springer.com/book/10.1007/978-3-642-31090-4

    work page doi:10.1007/978-3-642-31090-4 2013

  21. [21]

    Hom- L ie algebroids

    Camille Laurent-Gengoux and Joana Teles. Hom- L ie algebroids. Journal of Geometry and Physics , 68:69--75, 2013

    work page 2013

  22. [22]

    Universal higher Lie algebras of singular spaces and their symmetries

    Ruben Louis. Universal higher Lie algebras of singular spaces and their symmetries . PhD thesis, Universit \'e de Lorraine, 2022

    work page 2022

  23. [23]

    A series of Nash resolutions of a singular foliation

    Ruben Louis. A series of Nash resolutions of a singular foliation. Preprint, arXiv :2301.08706 [math. DG ], 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  24. [24]

    Left-symmetric algebroids

    Jiefeng Liu, Yunhe Sheng, Chengming Bai, and Zhiqi Chen. Left-symmetric algebroids. Mathematische Nachrichten , 289(14-15):1893--1908, 2016

    work page 1908

  25. [25]

    Kirill C. H. Mackenzie. General theory of L ie groupoids and L ie algebroids , volume 213 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 2005

    work page 2005

  26. [26]

    [Moo01] John Atwell Moody

    Omar Mohsen. Blow-up groupoid of singular foliations . Arxiv , 2021. https://arxiv.org/abs/2105.05201

    work page arXiv 2021

  27. [27]

    Tangent groupoid and tangent cones in sub-riemannian geometry, 2022

    Omar Mohsen. Tangent groupoid and tangent cones in sub-riemannian geometry, 2022

    work page 2022

  28. [28]

    On Lie algebroid actions and morphisms

    Tahar Mokri. On Lie algebroid actions and morphisms. Cahiers de Topologie et G\'eom\'etrie Diff\'erentielle Cat\'egoriques , 37(4):315--331, 1996

    work page 1996

  29. [29]

    Archana S. Morye. Note on the S erre- S wan theorem. Mathematische Nachrichten , 286(2-3):272--278, 2013

    work page 2013

  30. [30]

    T. Nagano. Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Japan , 18:398--404, 1966

    work page 1966

  31. [31]

    A. Nobile. Some properties of the Nash blowing-up. Pacific Journal of Mathematics , 60(1):297 -- 305, 1975

    work page 1975

  32. [32]

    Nash residues of singular holomorphic foliations

    Jean P aul B rasselet and T atsuo S uwa. Nash residues of singular holomorphic foliations. Asian Journal of Mathematics , 4:37--50, 2000

    work page 2000

  33. [33]

    Poisson (co)homology and isolated singularities

    Anne Pichereau. Poisson (co)homology and isolated singularities. Journal of Algebra , 299(2):747--777, 2006

    work page 2006

  34. [34]

    Almost L ie A lgebroids and C haracteristic C lasses

    Marcela Popescu and Paul Popescu. Almost L ie A lgebroids and C haracteristic C lasses. SIGMA , 15:021--12, 2019

    work page 2019

  35. [35]

    Orbits of families of vector fields on subcartesian spaces

    Jedrzej \'Sniatycki. Orbits of families of vector fields on subcartesian spaces. Annales de l'Institut Fourier , 53(7):2257--2296, 2003

    work page 2003

  36. [36]

    Residues of singular holomorphic foliations

    Sinan Sert\"oz. Residues of singular holomorphic foliations. Compositio Mathematica , 70(3):227--243, 1989

    work page 1989

  37. [37]

    P. Stefan. Accessibility and foliations with singularities. Bull. Am. Math. Soc. , 80:1142--1145, 1974

    work page 1974

  38. [38]

    P. Stefan. Integrability of systems of vector fields. J. Lond. Math. Soc., II. Ser. , 21:544--556, 1980

    work page 1980

  39. [39]

    Richard G. Swan. Vector B undles and P rojective modules. Transactions of the American Mathematical Society , 105(2):264--277, 1962. https://doi.org/10.1090/S0002-9947-1962-0143225-6

    work page doi:10.1090/s0002-9947-1962-0143225-6 1962

  40. [40]

    Id\' e aux de fonctions diff\' e rentiables

    Jean-Claude Tougeron. Id\' e aux de fonctions diff\' e rentiables. I . Ann. Inst. Fourier (Grenoble) , 18(fasc., fasc. 1):177--240, 1968

    work page 1968

  41. [41]

    A lagrangian for hamiltonian vector fields on singular poisson manifolds

    Yahya Turki. A lagrangian for hamiltonian vector fields on singular poisson manifolds. Journal of Geometry and Physics , 90:71--87, 2015

    work page 2015

  42. [42]

    Weinstein

    Pavol S evera and Alan J. Weinstein. Poisson geometry with a 3-form background. Progress of Theoretical Physics Supplement , 144:145--154, 2001

    work page 2001

  43. [43]

    Singular subalgebroids

    Marco Zambon. Singular subalgebroids. Annales de l'Institut Fourier , 72(6):2109--2190, 2022

    work page 2022

  44. [44]

    Hom- L ie algebroids and hom-left-symmetric algebroids

    Qingcheng Zhang, Haiyan Yu, and Chunyue Wang. Hom- L ie algebroids and hom-left-symmetric algebroids. Journal of Geometry and Physics , 116:187--203, 2017

    work page 2017