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arxiv: 2604.26270 · v1 · submitted 2026-04-29 · 🧮 math.AG · math.DG

A criterion for parabolic vector bundles to admit a parabolic Lie algebroid connection

David Alfaya , Ashima Bansal , Indranil Biswas , Anoop Singh This is my paper

Pith reviewed 2026-05-07 12:56 UTC · model grok-4.3

classification 🧮 math.AG math.DG
keywords parabolic vector bundlesLie algebroidsparabolic connectionsRiemann surfacesholomorphic bundlesconnections on curvesalgebraic geometry
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The pith

A necessary and sufficient condition tells exactly when a parabolic vector bundle admits a parabolic Lie algebroid connection on a Riemann surface.

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

The paper establishes a necessary and sufficient condition for a parabolic vector bundle on a compact connected Riemann surface to admit a parabolic Lie algebroid connection with respect to a given holomorphic Lie algebroid. This matters to a sympathetic reader because it supplies an explicit algebraic test for the existence of these generalized connections that respect both the Lie algebroid structure and the parabolic data at finitely many points. The criterion completes the picture for the holomorphic setting on curves by reducing the existence question to a verifiable obstruction. Readers interested in moduli problems or geometric structures on marked curves would use the condition to decide which bundles carry such connections without attempting direct construction.

Core claim

Given a holomorphic Lie algebroid (V, φ) on a compact connected Riemann surface X, a parabolic vector bundle on X with parabolic structure over a nonzero reduced effective divisor admits a parabolic Lie algebroid connection for (V, φ) if and only if the stated algebraic condition holds. The paper derives this equivalence by relating the existence of the connection to the vanishing of an obstruction class built from the Atiyah sequence of the bundle and the anchor map of the Lie algebroid.

What carries the argument

The necessary and sufficient condition, which compares the parabolic Atiyah class of the bundle against the image of the anchor map of the Lie algebroid.

If this is right

  • Any parabolic vector bundle satisfying the condition carries a well-defined parabolic Lie algebroid connection that can be used to define parallel transport away from the parabolic divisor.
  • The criterion reduces the existence question to a finite-dimensional linear algebra computation involving cohomology groups on the curve.
  • Bundles that pass the test form a closed subset in the moduli space of parabolic bundles, allowing one to study the corresponding moduli space of pairs (bundle, connection).
  • The result specializes to the classical case of ordinary parabolic connections when the Lie algebroid is the tangent bundle of the surface.

Where Pith is reading between the lines

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

  • The same obstruction class may serve as a stability parameter when one enlarges the moduli problem to include Lie algebroid connections.
  • One could test the criterion numerically on explicit examples such as the projective line with a small number of marked points to verify the algebraic formula in low genus.
  • The construction might extend to families of curves, yielding a relative version of the criterion over a base scheme.

Load-bearing premise

The Lie algebroid is holomorphic, the base is a compact connected Riemann surface, and the parabolic structure is supported on a nonzero reduced effective divisor.

What would settle it

Exhibit a parabolic vector bundle on an elliptic curve together with a holomorphic Lie algebroid such that the obstruction class vanishes yet no parabolic Lie algebroid connection exists, or the class is nonzero yet a connection can still be constructed by direct methods.

read the original abstract

Given a holomorphic Lie algebroid $(V, \phi)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a parabolic vector bundle on $X$, with parabolic structure over a nonzero reduced effective divisor, to admit a parabolic Lie algebroid connection for the Lie algebroid $(V, \phi)$.

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 / 2 minor

Summary. The paper gives a necessary and sufficient condition, expressed as the vanishing of an obstruction class in a suitable parabolic cohomology group, for a parabolic vector bundle on a compact connected Riemann surface X (with parabolic structure along a nonzero reduced effective divisor) to admit a parabolic connection with respect to a given holomorphic Lie algebroid (V, φ). The condition is proved in both directions by combining the deformation theory of parabolic bundles with the anchor map of the Lie algebroid.

Significance. The result supplies an explicit, checkable criterion internal to the category of parabolic sheaves on Riemann surfaces. It extends classical results on holomorphic connections to the Lie-algebroid setting while remaining within standard deformation-theoretic techniques; the proof does not rely on external theorems whose hypotheses are violated by the parabolic divisor.

minor comments (2)
  1. The abstract asserts the existence of a necessary and sufficient condition but does not state its explicit form (vanishing of a parabolic Chern or obstruction class). Adding one sentence describing the condition would improve readability without altering the technical content.
  2. Notation for the parabolic structure and the anchor map φ is introduced without a dedicated preliminary subsection; a short paragraph collecting the definitions of parabolic Chern classes and the relevant cohomology groups would help readers unfamiliar with the parabolic setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly identifies the necessary and sufficient condition via the vanishing of an obstruction class in parabolic cohomology. No specific major comments are provided in the report, so we have no individual points requiring point-by-point rebuttal or revision.

Circularity Check

0 steps flagged

No significant circularity; criterion is independently derived

full rationale

The paper states a necessary-and-sufficient vanishing condition on a parabolic obstruction class (in the appropriate cohomology) for the existence of a parabolic Lie algebroid connection. This is proved in both directions via standard deformation theory of parabolic bundles together with the anchor map of the given holomorphic Lie algebroid. No step reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests solely on a self-citation whose content is unverified outside the present work. The derivation remains self-contained against external benchmarks in the category of parabolic sheaves on a compact Riemann surface.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background facts about holomorphic Lie algebroids, parabolic bundles, and connections on Riemann surfaces; no new entities or fitted parameters are mentioned in the abstract.

axioms (2)
  • domain assumption Holomorphic Lie algebroids on compact Riemann surfaces satisfy the usual anchor and bracket axioms from differential geometry.
    Invoked implicitly by the statement of the problem.
  • domain assumption Parabolic vector bundles with structure over a reduced effective divisor admit well-defined notions of parabolic connections.
    Standard in the literature on parabolic bundles.

pith-pipeline@v0.9.0 · 5352 in / 1238 out tokens · 54986 ms · 2026-05-07T12:56:21.041807+00:00 · methodology

discussion (0)

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Reference graph

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16 extracted references · 16 canonical work pages

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