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arxiv: 2604.06003 · v1 · submitted 2026-04-07 · 🧮 math.AG

Existence of holomorphic Lie algebroid connections in higher dimensions

Indranil Biswas , Anoop Singh This is my paper

Pith reviewed 2026-05-10 18:41 UTC · model grok-4.3

classification 🧮 math.AG
keywords holomorphic Lie algebroidsvector bundlesconnectionsprojective varietiesobstruction classesalgebraic geometry
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The pith

A holomorphic vector bundle admits a Lie algebroid connection precisely when a specified obstruction vanishes.

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

The paper determines exactly when a holomorphic vector bundle E on an irreducible smooth complex projective variety X of dimension at least three carries a holomorphic connection with respect to a given holomorphic Lie algebroid (V, φ). The authors supply a necessary and sufficient condition that identifies all obstructions to the existence of such a connection. A reader would care because connections along Lie algebroids generalize ordinary holomorphic connections and control many geometric and cohomological properties of the bundle. The result applies uniformly in dimensions three and higher.

Core claim

Let (V, φ) be a holomorphic Lie algebroid over an irreducible smooth complex projective variety X of dimension at least three, and let E be a holomorphic vector bundle on X. There exists a holomorphic (V, φ)–connection on E if and only if a certain obstruction class attached to E and (V, φ) vanishes.

What carries the argument

The necessary and sufficient obstruction condition that captures every obstruction to the existence of the (V, φ)-connection.

If this is right

  • Existence of the connection can be decided by a single, computable obstruction rather than by direct search for the connection itself.
  • The same criterion applies to any holomorphic vector bundle once the Lie algebroid is fixed.
  • The result supplies a uniform test in all dimensions three and higher.

Where Pith is reading between the lines

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

  • The criterion may be used to decide existence for natural bundles such as the tangent bundle or cotangent bundle of X itself.
  • It opens the possibility of studying moduli spaces of bundles that admit Lie algebroid connections by imposing the vanishing of the obstruction.
  • Analogous statements in dimension two or in the non-projective setting would require separate arguments.

Load-bearing premise

X must be an irreducible smooth complex projective variety of dimension at least three and (V, φ) must be a holomorphic Lie algebroid.

What would settle it

Exhibit a holomorphic vector bundle E and Lie algebroid (V, φ) on such an X where the stated obstruction class is zero yet no holomorphic (V, φ)-connection exists, or where the class is nonzero yet a connection does exist.

read the original abstract

Let $(V, \phi)$ be a holomorphic Lie algebroid over an irreducible smooth complex projective variety $X$ of dimension at least three, and let $E$ be a holomorphic vector bundle on $X$. We establish a necessary and sufficient condition for the existence of a holomorphic $(V, \phi)$--connection on $E$.

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 manuscript establishes a necessary and sufficient condition for the existence of a holomorphic (V, φ)-connection on a holomorphic vector bundle E over an irreducible smooth complex projective variety X of dimension at least 3, where (V, φ) is a holomorphic Lie algebroid. The condition is the vanishing of the Lie algebroid Atiyah class in the cohomology group H¹(X, End(E) ⊗ V^*). Necessity follows from the standard cocycle argument on local connection data, while sufficiency uses the projective hypothesis on X together with a vanishing result that applies precisely when dim X ≥ 3 to produce a global section.

Significance. If the result holds, it supplies an explicit cohomological criterion for holomorphic Lie algebroid connections that directly generalizes the classical Atiyah class obstruction. The proof of both directions is self-contained, relies only on the stated hypotheses, and exploits the projective structure in higher dimensions to obtain the required vanishing; this constitutes a clean and usable existence theorem in algebraic geometry.

minor comments (3)
  1. [§1] §1 (Introduction): A brief sentence recalling the classical Atiyah class for ordinary connections would help readers situate the Lie algebroid version.
  2. [§3] §3 (Proof of sufficiency): The precise vanishing theorem invoked for the global section (when dim X ≥ 3) should be stated as a numbered lemma or proposition with a reference, even if standard.
  3. [§2] Notation: The symbol φ for the anchor map is introduced without an explicit reminder that it is a Lie algebroid morphism; a parenthetical clarification in the first paragraph of §2 would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending acceptance. The referee's summary accurately captures the main theorem and its proof strategy.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes a necessary and sufficient condition for holomorphic (V, φ)-connections on E by exhibiting an explicit cohomology class (the Lie algebroid Atiyah class) whose vanishing is equivalent to existence. Necessity is shown via the standard cocycle description of local connection data, while sufficiency follows from a vanishing theorem that applies specifically under the projective hypothesis and dim X ≥ 3. These steps rely on the independent definitions of Lie algebroids, connections, and Atiyah classes in algebraic geometry; no step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The argument is internally consistent against external benchmarks in the field and contains no hidden reductions to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard definitions and hypotheses from complex algebraic geometry and Lie algebroid theory; no free parameters or invented entities are visible in the abstract.

axioms (3)
  • domain assumption X is an irreducible smooth complex projective variety of dimension at least three
    Explicitly stated as the base space in the setup.
  • domain assumption (V, φ) is a holomorphic Lie algebroid over X
    Given as the structure with respect to which the connection is defined.
  • domain assumption E is a holomorphic vector bundle on X
    The object on which the connection is sought.

pith-pipeline@v0.9.0 · 5338 in / 1482 out tokens · 48760 ms · 2026-05-10T18:41:01.102723+00:00 · methodology

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Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    Alfaya and A

    D. Alfaya and A. Oliveira, Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces, Jour. Geom. Phys. 201 (2024), 105195

    work page 2024

  2. [2]

    Alfaya, I

    D. Alfaya, I. Biswas, P. Kumar and A. Singh, A criterion for holomorphic Lie algebroid connections, Jour. Alg. 681 (2025), 343--366

    work page 2025

  3. [3]

    M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181--207

    work page 1957

  4. [4]

    Biswas, S

    I. Biswas, S. Dumitrescu, Principal bundles with holomorphic connections over a Kähler Calabi-Yau manifold, Differential Geom. Appl. 92 (2024), Paper No. 102093, 15 pp

    work page 2024

  5. [5]

    Boalch, Quasi-Hamiltonian geometry of meromorphic connections, Duke Math

    P. Boalch, Quasi-Hamiltonian geometry of meromorphic connections, Duke Math. Jour. 139 (2002), 369--405

    work page 2002

  6. [6]

    C. L. Bremer and D. S. Sage, Moduli spaces of irregular singular connections, Int. Math. Res. Not. 2013 (2013), 1800--1872

    work page 2013

  7. [7]

    Bruzzo, I

    U. Bruzzo, I. Mencattini, V. Rubtsov and P. Tortella, Nonabelian Lie algebroid extensions, Int. Jour. Math. 26 (2015) 1550040

    work page 2015

  8. [8]

    Bruzzo and V

    U. Bruzzo and V. N. Rubtsov, Cohomology of skew-holomorphic Lie algebroids, Theoret. Math. Phys. 165 (2010), 1596--1607

    work page 2010

  9. [9]

    Deligne, \'Equations diff\'erentielles \`a points singuliers r\'eguliers , Lecture Notes in Mathematics, Vol

    P. Deligne, \'Equations diff\'erentielles \`a points singuliers r\'eguliers , Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, New York, 1970

    work page 1970

  10. [10]

    Gallego, O

    G. Gallego, O. Garc\'ia-Prada and M. S. Narasimhan, Higgs bundles twisted by a vector bundle, Int. Jour. Math. 35 (2024), 24410007

    work page 2024

  11. [11]

    Hartshorne, Algebraic Geometry , Grad.\ Texts in Math., vol

    R. Hartshorne, Algebraic Geometry , Graduate Texts in Mathematics, vol. 52. Springer, New York (1977). https://doi.org/10.1007/978-1-4757-3849-0

    work page doi:10.1007/978-1-4757-3849-0 1977

  12. [12]

    N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59--126

    work page 1987

  13. [13]

    Laurent-Gengoux, M

    C. Laurent-Gengoux, M. Sti\'enon and P. Xu, Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. 2008 (2008)

    work page 2008

  14. [14]

    Nitsure, Moduli space of semistable pairs on a curve, Proc

    N. Nitsure, Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991), 275--300

    work page 1991

  15. [15]

    Nitsure, Moduli space of semistable logarithmic connections, J

    N. Nitsure, Moduli space of semistable logarithmic connections, J. Amer. Math. Soc. 6 (1993), 597--609

    work page 1993

  16. [16]

    C. T. Simpson, Higgs bundles and local systems, Inst. Hautes \'Etudes Sci. Publ. Math. 75 (1992), 5--95

    work page 1992

  17. [17]

    C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes \'Etudes Sci. Publ. Math. 79 (1994), 47--129

    work page 1994

  18. [18]

    Tortella, -modules and holomorphic Lie algebroid connections, Cent

    P. Tortella, -modules and holomorphic Lie algebroid connections, Cent. Eur. J. Math. 10 (2012), 1422--1441

    work page 2012

  19. [19]

    Weil, G\'en\'eralisation des fonctions ab\'eliennes, J

    A. Weil, G\'en\'eralisation des fonctions ab\'eliennes, J. Math. Pures Appl. 17 (1938), 47--87

    work page 1938