Direct image and pullback of Parabolic vector bundles

Chandranandan Gangopadhyay; Indranil Biswas

arxiv: 2604.05411 · v1 · submitted 2026-04-07 · 🧮 math.AG

Direct image and pullback of Parabolic vector bundles

Indranil Biswas , Chandranandan Gangopadhyay This is my paper

Pith reviewed 2026-05-10 19:27 UTC · model grok-4.3

classification 🧮 math.AG

keywords parabolic vector bundlesroot stacksdirect imagepullbackvector bundlescurvesalgebraic geometry

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The pith

Direct image and pullback operations on parabolic vector bundles match the corresponding operations on vector bundles over root stacks.

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

The paper takes as given an equivalence, due to Niels Borne, between parabolic vector bundles on a curve and ordinary vector bundles on an associated root stack. It then proves that the direct-image and pullback functors defined for parabolic bundles are transported exactly to the ordinary direct-image and pullback functors on the root-stack side. Because the two categories are identified by the equivalence, any property or computation that can be performed with one set of operations immediately translates to the other. The authors also record some concrete applications that follow from this identification.

Core claim

We show that these two notions correspond to the notions direct image of vector bundles on root stacks and pullback of vector bundles on root stacks respectively.

What carries the argument

The natural equivalence between parabolic vector bundles on curves and vector bundles on the corresponding root stacks.

Load-bearing premise

The equivalence between parabolic vector bundles on curves and ordinary vector bundles on root stacks, established by Niels Borne, is taken as given.

What would settle it

An explicit parabolic vector bundle on a curve together with a morphism to the base curve such that the parabolic direct image fails to correspond, under Borne's equivalence, to the ordinary direct image of the associated bundle on the root stack.

read the original abstract

Niels Borne established a natural correspondence between the parabolic vector bundles on curves and vector bundles on root stacks. The notions of direct image of parabolic vector bundles and pullback of parabolic vector bundles were studied in \cite{Alfaya_Biswas}. We show that these two notions correspond to the notions direct image of vector bundles on root stacks and pullback of vector bundles on root stacks respectively. Some applications of this correspondence are given.

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

This paper checks that direct image and pullback for parabolic bundles match the root-stack versions under Borne's equivalence, a standard but needed compatibility step.

read the letter

The core of the paper is showing that the direct image and pullback operations defined for parabolic vector bundles on curves correspond exactly to the direct image and pullback of vector bundles on the associated root stacks, once you use Borne's equivalence of categories. That is the main claim, and it follows directly from the setup in the cited Alfaya-Biswas work on those operations plus Borne's original correspondence. It is new in the narrow sense that this specific functoriality was not already written down, though it is the sort of check one expects once the two frameworks are in place. The authors also list some applications, which presumably let people transfer known results from one side to the other without redoing the work. That is the useful part for people already inside this subfield. The argument looks solid on the surface because it rests on two established references rather than introducing new definitions or heavy machinery. The proofs are likely diagram chases or direct verifications of the universal properties, which is routine but necessary. I do not see load-bearing gaps from the abstract, though the usual assumptions on the base field, the rationality of weights, and the curve being smooth are probably carried over from the priors. If those are handled cleanly, the result holds. This is aimed at specialists who work with parabolic bundles or root stacks on curves and want to switch languages freely. A reader outside that niche will not find much to use. It is the kind of technical note that deserves a serious referee because the claim is precise, the references are appropriate, and the area still cites such compatibilities. I would send it to review rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes that the direct image and pullback operations on parabolic vector bundles, as defined in Alfaya-Biswas, correspond exactly to the direct image and pullback operations on vector bundles over root stacks under the equivalence of categories constructed by Niels Borne. Applications of the resulting identification are indicated.

Significance. If the stated compatibility holds, the result supplies a useful bridge between two presentations of the same category, allowing techniques developed in one setting (e.g., root-stack cohomology or stability) to be transferred directly to the other. The paper correctly positions itself as a functoriality check rather than a new foundational construction and credits the prior references appropriately.

minor comments (2)

The abstract and introduction should explicitly record the standing assumptions on the base curve (smooth projective, over an algebraically closed field of characteristic zero, etc.) and on the parabolic weights, since these hypotheses are inherited from Borne and Alfaya-Biswas and are needed to invoke the equivalence.
In the applications section, state precisely which new statements become immediate from the correspondence (e.g., a specific vanishing or stability result) rather than leaving the benefit implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report, which accurately summarizes the main result of the manuscript: the identification of direct image and pullback functors for parabolic vector bundles (as defined in Alfaya-Biswas) with the corresponding functors for vector bundles on root stacks, via Borne's equivalence. We appreciate the positive assessment of the paper's significance as a functoriality check and the recommendation for minor revision.

Circularity Check

0 steps flagged

Minor self-citation to prior definitions; central compatibility claim is independent

full rationale

The paper takes Borne's external equivalence of categories as given and verifies that the direct image/pullback operations (defined in the cited Alfaya-Biswas work) are compatible with the corresponding operations on root stacks. This is a standard functoriality check once the equivalence is fixed. The only self-citation is to the source of the parabolic operations being compared, which does not reduce the claimed result to a tautology or to quantities defined only within this paper. No load-bearing step collapses by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the prior Borne correspondence and on the definitions of direct image and pullback given in the cited Alfaya-Biswas paper; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption Niels Borne's natural correspondence between parabolic vector bundles on curves and vector bundles on root stacks
The paper invokes this established equivalence as the starting point for showing that the two operations correspond.

pith-pipeline@v0.9.0 · 5357 in / 1198 out tokens · 54224 ms · 2026-05-10T19:27:32.361105+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show that these two notions correspond to the notions direct image of vector bundles on root stacks and pullback of vector bundles on root stacks respectively. (Theorem 3.1)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Niels Borne established a natural correspondence between the parabolic vector bundles on curves and vector bundles on root stacks.

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.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Alfaya and I

D. Alfaya and I. Biswas, Pullback and direct image of parabolic connections and parabolic H iggs bundles, Int. Math. Res. Not. 22 (2023), 19546--19591

work page 2023
[2]

Alfaya, I

D. Alfaya, I. Biswas and F.-X. Machu, Pullback and direct image of parabolic Higgs bundles and parabolic connections with symplectic and orthogonal structures, Illinois Jour. Math. (to appear)

work page
[3]

Borne and A

N. Borne and A. Laaroussi, Parabolic connections and stack of roots, Bull. Sci. Math. , 187:Paper No. 103294, 33, 2023

work page 2023
[4]

Borne, Fibr\'es paraboliques et champ des racines, Int

N. Borne, Fibr\'es paraboliques et champ des racines, Int. Math. Res. Not. 16 , Art. ID rnm049, 38, 2007

work page 2007
[5]

Cadman, Using stacks to impose tangency conditions on curves, Amer

C. Cadman, Using stacks to impose tangency conditions on curves, Amer. Jour. Math. 129 (2007), 405--427

work page 2007
[6]

Chakraborty and S

S. Chakraborty and S. Majumder, Orthogonal and symplectic parabolic connections and stack of roots, Bull. Sci. Math. 191 :103397, 2024

work page 2024
[7]

Olsson, Algebraic spaces and stacks , volume 62 of American Mathematical Society Colloquium Publications

M. Olsson, Algebraic spaces and stacks , volume 62 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, 2016

work page 2016
[8]

V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205--239

work page 1980

[1] [1]

Alfaya and I

D. Alfaya and I. Biswas, Pullback and direct image of parabolic connections and parabolic H iggs bundles, Int. Math. Res. Not. 22 (2023), 19546--19591

work page 2023

[2] [2]

Alfaya, I

D. Alfaya, I. Biswas and F.-X. Machu, Pullback and direct image of parabolic Higgs bundles and parabolic connections with symplectic and orthogonal structures, Illinois Jour. Math. (to appear)

work page

[3] [3]

Borne and A

N. Borne and A. Laaroussi, Parabolic connections and stack of roots, Bull. Sci. Math. , 187:Paper No. 103294, 33, 2023

work page 2023

[4] [4]

Borne, Fibr\'es paraboliques et champ des racines, Int

N. Borne, Fibr\'es paraboliques et champ des racines, Int. Math. Res. Not. 16 , Art. ID rnm049, 38, 2007

work page 2007

[5] [5]

Cadman, Using stacks to impose tangency conditions on curves, Amer

C. Cadman, Using stacks to impose tangency conditions on curves, Amer. Jour. Math. 129 (2007), 405--427

work page 2007

[6] [6]

Chakraborty and S

S. Chakraborty and S. Majumder, Orthogonal and symplectic parabolic connections and stack of roots, Bull. Sci. Math. 191 :103397, 2024

work page 2024

[7] [7]

Olsson, Algebraic spaces and stacks , volume 62 of American Mathematical Society Colloquium Publications

M. Olsson, Algebraic spaces and stacks , volume 62 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, 2016

work page 2016

[8] [8]

V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205--239

work page 1980