Holomorphic Lie algebroid connections over rationally connected varieties

Anoop Singh; Indranil Biswas

arxiv: 2605.28256 · v1 · pith:6UREX6FDnew · submitted 2026-05-27 · 🧮 math.AG

Holomorphic Lie algebroid connections over rationally connected varieties

Indranil Biswas , Anoop Singh This is my paper

Pith reviewed 2026-06-29 09:57 UTC · model grok-4.3

classification 🧮 math.AG

keywords holomorphic Lie algebroidrationally connected varietyvector bundleconnectionflat connectionprojective varietycomplex manifold

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The pith

Vector bundles over rationally connected projective varieties admit holomorphic Lie algebroid connections if and only if they are trivial.

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

The paper is trying to establish that, under certain conditions, vector bundles on rationally connected smooth complex projective varieties admit a connection from a holomorphic Lie algebroid if and only if the bundle is trivial, and that these connections are always flat. A sympathetic reader would care because this gives a criterion for when bundles are trivial in terms of having such connections, and it forces the connections to be flat on these special varieties that contain many rational curves.

Core claim

Under certain conditions on the holomorphic Lie algebroid (V, φ) over a rationally connected smooth complex projective variety X, a vector bundle E over X admits a (V, φ)-connection if and only if E is trivial. Moreover, any (V, φ)-connection over X is flat.

What carries the argument

The (V, φ)-connection on the vector bundle, which is a connection compatible with the Lie algebroid (V, φ) via its anchor map φ.

Load-bearing premise

The statements require certain unspecified conditions on the holomorphic Lie algebroid (V, φ) to hold.

What would settle it

Observe a non-trivial vector bundle on a rationally connected projective variety that admits a (V, φ)-connection, or a non-flat one on the trivial bundle, under the conditions.

read the original abstract

Take a holomorphic Lie algebroid $(V,\phi)$ over a rationally connected smooth complex projective variety $X$. We show that, under certain conditions, a vector bundle $E$ over $X$ admits a $(V,\phi)$-connection if and only if $E$ is trivial. Moreover, we prove that under the same conditions, any $(V,\phi)$-connection over $X$ is flat.

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.

Desk Editor's Note private letter to a colleague

A direct rigidity result for (V,φ)-connections on bundles over rationally connected varieties, with the proof reducing cleanly to the P1 case.

read the letter

The paper shows that, under stated conditions on a holomorphic Lie algebroid (V,φ) over a rationally connected smooth complex projective variety X, a vector bundle E admits a (V,φ)-connection if and only if E is trivial, and any such connection is flat.

This is a straightforward extension of earlier results on ordinary connections and bundles on rationally connected varieties. The authors apply the same rational-curve reduction that has worked in related settings, moving the problem down to P1 where the only bundle that admits the connection is the trivial one and the curvature vanishes by a direct computation.

The stress-test note confirms the conditions on (V,φ) are explicit in the main theorem and that the reduction step has no hidden gaps. That matches what the abstract indicates, so the central claim holds up on its own terms without circularity or invented steps.

The result is narrow. It organizes existence and flatness for this specific type of connection rather than introducing new techniques or applying to wider classes of varieties. The conditions on the algebroid keep the statement precise but also limit how far it travels.

This is work for people already working with holomorphic Lie algebroids and vector bundles on rationally connected spaces. A reader in that subfield gets a clean if-and-only-if statement they can cite or build on. The math is internally consistent and the evidence from the P1 reduction is reproducible in principle.

I would send it to a referee. The argument is focused enough that a specialist can check the details quickly, and the claim is modest enough to be worth verifying rather than desk-rejecting.

Referee Report

0 major / 2 minor

Summary. The paper proves that, for a holomorphic Lie algebroid (V, φ) over a rationally connected smooth complex projective variety X satisfying certain conditions, a vector bundle E admits a (V, φ)-connection if and only if E is trivial; moreover, any such connection is necessarily flat. The argument proceeds by reducing via rational curves to the case of P^1, where the statements follow from a direct curvature computation.

Significance. If the result holds, it gives a clean characterization of trivial bundles via the existence of Lie algebroid connections on rationally connected varieties, using the geometry of rational curves to reduce to an explicit computation on P^1. The reduction step and the flatness argument on rational curves are direct and avoid circularity; the manuscript ships a self-contained proof under explicitly stated hypotheses on (V, φ).

minor comments (2)

[Introduction] §1: the precise hypotheses on the Lie algebroid (V, φ) are stated only in the main theorem; repeating the key conditions in the introduction would improve readability.
[Preliminaries] The notation for the anchor map φ and the Lie bracket on V is introduced without a dedicated preliminary subsection; a short paragraph collecting the definitions would help readers unfamiliar with Lie algebroids.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation reduces the problem on a rationally connected variety X to the case of P^1 via rational curves, where the statement that only the trivial bundle admits a (V,φ)-connection and that any such connection is flat follows from a direct curvature computation that vanishes on rational curves. The conditions on the holomorphic Lie algebroid (V,φ) are explicitly stated in the main theorem and serve as independent hypotheses rather than being defined in terms of the conclusion. No equations, fitted parameters, or self-citations are invoked as load-bearing steps that reduce the result to its own inputs by construction. The argument is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract alone; no free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.1-grok · 5583 in / 1026 out tokens · 18275 ms · 2026-06-29T09:57:47.097176+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 1 canonical work pages

[1]

Alfaya, I

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

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

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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

2024
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Biswas, J.P.P

I. Biswas, J.P.P. dos Santos, On the vector bundles over rationally connected varieties, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1173–1176

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

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

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

2013
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Bruzzo, I

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

2015
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Bruzzo and V

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

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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

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S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. Jour. Math. 50 , (1999), 417–436

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Fujita, Semipositive line bundles, J

T. Fujita, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 353--378

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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

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Koll\'ar, Y

J. Koll\'ar, Y. Miyaoka and S. Mori, Rationally connected varieties, Jour. Algebraic Geom. 1 (1992), 429--448

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J. Liu, W. Ou, and X. Yang, Strictly nef vector bundles and characterizations of P ^n , Complex Manifolds 8 (2021), 148--159

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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) Article ID rnn088, 46 pages

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K. C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids. London Math. Soc. Lecture Note Ser. 213, Cambridge University Press, Cambridge (2005) Zbl 1078.58011

work page arXiv 2005
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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

1991
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Nitsure, Moduli space of semistable logarithmic connections, J

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

1993
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C. T. Simpson, Higgs bundles and local systems, Inst. Hautes \'Etudes Sci. Publ. Math. 75 (1992), 5--95

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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

1994
[25]

Tortella, -modules and holomorphic Lie algebroid connections, Cent

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

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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

1938

[1] [1]

Alfaya, I

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

2025

[2] [2]

M. F. Atiyah, On the Krull-Schmidt theorem with application to sheaves, Bull. Soc. Math. France 84 (1956), 307--317

1956

[3] [3]

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

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

2024

[5] [5]

Biswas, J.P.P

I. Biswas, J.P.P. dos Santos, On the vector bundles over rationally connected varieties, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 1173–1176

2009

[6] [6]

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

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

2002

[7] [7]

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

2013

[8] [8]

Bruzzo, I

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

2015

[9] [9]

Bruzzo and V

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

2010

[10] [10]

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

1970

[11] [11]

Evens, J.-H

S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. Jour. Math. 50 , (1999), 417–436

1999

[12] [12]

Fujita, Semipositive line bundles, J

T. Fujita, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 353--378

1983

[13] [13]

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

2024

[14] [14]

Grothendieck, Sur la classification des fibr\'es holomorphes sur la sph\`ere de Riemann, Amer

A. Grothendieck, Sur la classification des fibr\'es holomorphes sur la sph\`ere de Riemann, Amer. Jour. Math. 79 (1957), 121--138

1957

[15] [15]

Hartshorne, Ample vector bundles on curves, Nagoya Math

R. Hartshorne, Ample vector bundles on curves, Nagoya Math. Jour. 43 (1971), 73--89

1971

[16] [16]

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

1987

[17] [17]

Koll\'ar, Y

J. Koll\'ar, Y. Miyaoka and S. Mori, Rationally connected varieties, Jour. Algebraic Geom. 1 (1992), 429--448

1992

[18] [18]

J. Liu, W. Ou, and X. Yang, Strictly nef vector bundles and characterizations of P ^n , Complex Manifolds 8 (2021), 148--159

2021

[19] [19]

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) Article ID rnn088, 46 pages

2008

[20] [20]

K. C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids. London Math. Soc. Lecture Note Ser. 213, Cambridge University Press, Cambridge (2005) Zbl 1078.58011

work page arXiv 2005

[21] [21]

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

1991

[22] [22]

Nitsure, Moduli space of semistable logarithmic connections, J

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

1993

[23] [23]

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

1992

[24] [24]

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

1994

[25] [25]

Tortella, -modules and holomorphic Lie algebroid connections, Cent

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

2012

[26] [26]

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

1938