The Dolbeault geometric Langlands correspondence for type A groups beyond the elliptic locus

Yukinobu Toda

arxiv: 2606.28878 · v1 · pith:QSYEAPGJnew · submitted 2026-06-27 · 🧮 math.AG · math.RT

The Dolbeault geometric Langlands correspondence for type A groups beyond the elliptic locus

Yukinobu Toda This is my paper

Pith reviewed 2026-06-30 08:29 UTC · model grok-4.3

classification 🧮 math.AG math.RT

keywords geometric LanglandsDolbeault correspondenceHitchin fibrationspectral curvestype A singularitiesHiggs bundlesWhittaker normalizationcategorical DT theory

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The pith

A Dolbeault geometric Langlands equivalence holds for GL_r and SL_r/PGL_r over the locus of spectral curves with at worst type A singularities.

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

This paper proves that the Dolbeault geometric Langlands equivalence for GL_r and SL_r/PGL_r extends to an open set in the Hitchin base that includes spectral curves with at worst type A singularities. The equivalence connects coherent sheaves on the moduli of semistable Higgs bundles to a limit category built from the full, non-quasi-compact Higgs moduli stack. The central technical step is establishing the Whittaker normalization conjecture on this larger locus by building on the GL_2 proof strategy. As a result, the conjecture also holds for SL_2/PGL_2 over the reduced spectral curve locus.

Core claim

The Dolbeault geometric Langlands equivalence, which links categorical Donaldson-Thomas theory to the geometric Langlands correspondence, is shown to hold for GL_r and the dual pair SL_r/PGL_r over the open locus of the Hitchin base corresponding to spectral curves with at worst type A singularities, with no restriction on the number of irreducible components. This is achieved by proving the Whittaker normalization conjecture in this setting, extending the approach used previously for GL_2 over reduced curves. The use of limit categories is key to handling the infinitely many Harder-Narasimhan strata in the full moduli stack beyond the elliptic locus.

What carries the argument

The Whittaker normalization conjecture over the locus of spectral curves with type A singularities, which enables the equivalence between coherent sheaves on semistable Higgs moduli stacks and limit categories of the full Higgs stack.

If this is right

The equivalence holds over a strictly larger open locus than the elliptic one.
It applies to spectral curves with any number of irreducible components as long as singularities are at worst type A.
The result yields the Dolbeault conjecture for SL_2/PGL_2 over the reduced spectral curve locus as a consequence.
The correspondence relates categories from categorical Donaldson-Thomas theory to geometric Langlands.

Where Pith is reading between the lines

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

This suggests that proving similar normalization conjectures for other singularity types could extend the correspondence even further.
The approach may help address challenges with non-quasi-compact moduli stacks in other parts of the geometric Langlands program.
Explicit computations for low-rank cases with type A singularities could provide checks on the equivalence.

Load-bearing premise

The Whittaker normalization conjecture holds over the locus of spectral curves with type A singularities and the strategy from the GL_2 case extends without new obstructions.

What would settle it

An explicit counterexample where the Whittaker normalization fails for a specific spectral curve with a type A singularity, such as a cusp or node, would disprove the equivalence.

Figures

Figures reproduced from arXiv: 2606.28878 by Yukinobu Toda.

**Figure 1.** Figure 1: A schematic comparison between the elliptic locus and the reduced locus. On the elliptic locus, the spectral curve Cb is integral. On the reduced locus, the spectral curve may be reducible; the right-hand picture illustrates a reduced spectral curve whose smooth irreducible components meet transversely. By the first remark above, as a corollary, we obtain the following result, which gives a version of the … view at source ↗

**Figure 2.** Figure 2: Flowchart of the proof of Theorem 1.5. 1.6. Acknowledgements. The author would like to thank Tudor P˘adurariu and Tasuki Kinjo for useful discussions, and Yongbin Ruan and Yaoxiong Wen for the discussions and comments during the Mini-Workshop on Topological Geometric Langlands held on April 25–26, 2026. The author is supported by the World Premier International Research Center Initiative (WPI Initiative),… view at source ↗

read the original abstract

In this paper, we prove a Dolbeault geometric Langlands equivalence for $\GL_r$ and for the Langlands dual pair $\SL_r/\PGL_r$ over an open locus of the Hitchin base which strictly contains the elliptic locus. This open locus contains the points corresponding to spectral curves with at worst type $A$ singularities, without any restriction on the number of irreducible components. The Dolbeault geometric Langlands equivalence considered here is the one formulated in our previous work with Tudor P\u{a}durariu, which links categorical Donaldson--Thomas theory with the geometric Langlands correspondence. It relates coherent sheaves on moduli stacks of semistable Higgs bundles to the limit category associated with the full moduli stack of Higgs bundles. The use of limit categories is essential beyond the elliptic locus, where the full Higgs moduli stack is no longer quasi-compact and contains infinitely many Harder--Narasimhan strata. The key step is to prove the Whittaker normalization conjecture over the locus of spectral curves with type $A$ singularities, following and extending the strategy developed in the author's proof of the $\GL_2$ case over the reduced spectral curve locus. As a consequence, we also obtain the Dolbeault geometric Langlands conjecture for $\SL_2/\PGL_2$ over the reduced spectral curve locus.

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

Toda extends the Dolbeault geometric Langlands equivalence to type A singularities for GL_r by proving Whittaker normalization there, building directly on the Pădurariu joint work and the GL2 case.

read the letter

The key advance is a proof of the Dolbeault geometric Langlands equivalence for GL_r and SL_r/PGL_r over an open set in the Hitchin base that includes spectral curves with type A singularities of arbitrary numbers of components. This strictly enlarges the verified domain past the elliptic locus. The paper uses the limit-category formulation from the earlier joint work to handle the non-quasi-compact moduli stack, then establishes the Whittaker normalization conjecture on this larger locus by adapting the strategy from the author's GL2 result over reduced curves. As a side outcome it also settles the SL2/PGL2 case on the reduced locus.

The approach looks consistent with the prior papers it cites. The use of limit categories is a natural way to move beyond quasi-compactness, and extending the normalization step to type A singularities without a component restriction is the concrete new step.

The main limitation is that the abstract gives no explicit derivation or error controls for how the GL2 strategy lifts to higher rank without fresh obstructions. The dependence on the Pădurariu formulation and the self-cited GL2 work is explicit, but the soundness of the extension cannot be checked from the given text alone. No internal contradictions appear in the stated claims.

This is specialized work aimed at researchers already working in categorical geometric Langlands and categorical Donaldson-Thomas theory. Someone familiar with the elliptic case and the GL2 paper will see the incremental value immediately. It is worth sending to a serious referee because the claimed enlargement of the locus is precise and the technical step is isolated, even though the details will need close checking.

Referee Report

1 major / 0 minor

Summary. The manuscript proves a Dolbeault geometric Langlands equivalence for GL_r and the dual pair SL_r/PGL_r over an open locus of the Hitchin base strictly larger than the elliptic locus. This locus includes all spectral curves with at worst type A singularities (no restriction on the number of irreducible components). The equivalence is the one formulated in prior joint work with Tudor Pădurariu, relating coherent sheaves on moduli stacks of semistable Higgs bundles to a limit category of the full Higgs moduli stack; the key technical step is a proof of the Whittaker normalization conjecture over the type A locus obtained by extending the strategy from the author's earlier GL_2 result over the reduced spectral curve locus. As a corollary the result also yields the conjecture for SL_2/PGL_2 over the reduced locus.

Significance. If the central extension holds, the result would meaningfully enlarge the known range of the Dolbeault geometric Langlands correspondence to higher-rank groups and to spectral curves with controlled singularities, while introducing limit categories to manage the non-quasi-compactness and infinitely many Harder-Narasimhan strata that appear beyond the elliptic locus. The absence of any restriction on the number of irreducible components is a concrete strengthening over previous statements.

major comments (1)

[Abstract] Abstract and the paragraph describing the key step: the claim that the GL_2 strategy extends to higher rank over the type A locus without new obstructions is load-bearing for the central theorem, yet the provided text supplies no explicit verification that the adaptation of the Whittaker normalization argument carries over when the spectral curve may have multiple components or when r > 2; a dedicated subsection outlining the new technical points (or confirming their absence) is required.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the referee's thoughtful comments on our manuscript. We are grateful for the recognition of the significance of extending the Dolbeault geometric Langlands correspondence beyond the elliptic locus. We respond to the major comment as follows.

read point-by-point responses

Referee: [Abstract] Abstract and the paragraph describing the key step: the claim that the GL_2 strategy extends to higher rank over the type A locus without new obstructions is load-bearing for the central theorem, yet the provided text supplies no explicit verification that the adaptation of the Whittaker normalization argument carries over when the spectral curve may have multiple components or when r > 2; a dedicated subsection outlining the new technical points (or confirming their absence) is required.

Authors: We agree that an explicit outline of the adaptations would improve the clarity of the manuscript. While the body of the paper details the proof by extending the GL_2 arguments using the type A singularity assumption, which ensures that the relevant cohomology groups and deformation spaces behave similarly regardless of the number of components or rank r (as the local models are uniform), we will add a new subsection in the introduction following the statement of the main theorem. This subsection will summarize the key technical points from the GL_2 case and confirm their direct applicability here, noting the absence of new obstructions due to the controlled nature of type A singularities. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation consists of a new proof of the Whittaker normalization conjecture over the type A locus by extending a prior strategy from the GL_2 case, together with a formulation from joint prior work. This constitutes independent mathematical content rather than any reduction of the output to inputs by definition, fitting, or self-citation chain. No equations, ansatzes, or uniqueness claims are shown to collapse the result to prior fitted quantities or unverified self-references. The work is self-contained as a standard mathematical extension with externally verifiable proof steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a proof in algebraic geometry and category theory; it inherits standard background axioms from prior literature on moduli stacks and Higgs bundles. No free parameters or invented entities are visible in the abstract. The central claim rests on the correctness of the Whittaker normalization conjecture (treated as a domain assumption to be proved) and on the formulation from the cited prior joint work.

axioms (2)

domain assumption The Dolbeault geometric Langlands equivalence is the one formulated in prior joint work with Tudor Pădurariu linking categorical Donaldson-Thomas theory to the geometric Langlands correspondence.
Invoked in the abstract as the equivalence being proved; the paper does not re-derive the formulation.
domain assumption The full moduli stack of Higgs bundles is no longer quasi-compact beyond the elliptic locus and contains infinitely many Harder-Narasimhan strata, necessitating the use of limit categories.
Stated as the reason limit categories are essential; treated as background from the theory of moduli stacks.

pith-pipeline@v0.9.1-grok · 5767 in / 1687 out tokens · 34793 ms · 2026-06-30T08:29:41.844288+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 9 canonical work pages · 1 internal anchor

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A proof of Dolbeault geometric Langlands for $\mathrm{GL}_2$ with reduced spectral curves

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Arinkin, D

[ABC+a] D. Arinkin, D. Beraldo, J. Campbell, L. Chen, J. Faergeman, D. Gaitsgory, K. Lin, S. Raskin, and N. Rozenblyum,Proof of the geometric Langlands conjecture II: Kac- Moody localization and the FLE, arXiv:2405.03648. [ABC+b] D. Arinkin, D. Beraldo, L. Chen, J. Faergeman, D. Gaitsgory, K. Lin, S. Raskin, and N. Rozenblyum,Proof of the geometric Langla...

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Ginzburg,The global nilpotent variety is Lagrangian, Duke Mathematical Journal 109(2001), no

[Gin01] V. Ginzburg,The global nilpotent variety is Lagrangian, Duke Mathematical Journal 109(2001), no. 3, 511–519. [GLS07] Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin,Introduction to singulari- ties and deformations, Springer Monographs in Mathematics, Springer, Berlin,

2001

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Gaitsgory and S

[GRa] D. Gaitsgory and S. Raskin,Proof of the geometric Langlands conjecture I: construction of the functor, arXiv:2405.03599. [GRb] ,Proof of the geometric Langlands conjecture V: the multiplicity one theorem, arXiv:2409.09856. [GR17] D. Gaitsgory and N. Rozenblyum,A study in derived algebraic geometry. Vol. I. Cor- respondences and duality, Mathematical...

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Groechenig and S

[GS22] M. Groechenig and S. Shen,Complex K-theory of moduli spaces of Higgs bundles, arXiv:2212.10695,

work page arXiv

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Hitchin,Stable bundles and integrable systems, Duke Math

[Hit87] N. Hitchin,Stable bundles and integrable systems, Duke Math. J.54(1987), no. 1, 91–114. [HL] D. Halpern-Leistner,On the structure of instability in moduli theory, arXiv:1411.0627. [HL97] D. Huybrechts and M. Lehn,Geometry of moduli spaces of sheaves, Aspects in Mathe- matics, vol. E31, Vieweg,

work page arXiv 1987

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Halpern-Leistner,DerivedΘ-stratifications and theD-equivalence conjecture, arXiv:2010.01127, to appear in Ann

[HL20] D. Halpern-Leistner,DerivedΘ-stratifications and theD-equivalence conjecture, arXiv:2010.01127, to appear in Ann. of Math.,

work page arXiv 2010

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Halpern-Leistner and S

[HLS20] D. Halpern-Leistner and S. V. Sam,Combinatorial constructions of derived equivalences, J. Amer. Math. Soc.33(2020), no. 3, 735–773. [Kha21] A. Khan,Dimensional classicality criterion for derived stacks, Preprint, available at https://www.preschema.com/papers/classicality.pdf,

2020

[8] [8]

Kouvidakis and T

[KP95] A. Kouvidakis and T. Pantev,The automorphism group of the moduli space of semi stable bundles, Mathematische Annalen302(1995), 225–268. [Li21] M. Li,Construction of Poincar´ e sheaf on stack of rank 2 Higgs bundles, Adv. Math. 376(2021), Paper No. 107437,

1995

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Lieblich,Moduli of twisted sheaves, Duke Math

[Lie07] M. Lieblich,Moduli of twisted sheaves, Duke Math. J.138(2007), no. 1, 23–118. [MRV17] M. Melo, A. Rapagnetta, and F. Viviani,Fine compactified Jacobians of reduced curves, Trans. Amer. Math. Soc.369(2017), no. 8, 5341–5402. [MRV19] ,Fourier-Mukai and autoduality for compactified Jacobians. II, Geometry and Topology23(2019), 2335–2395. [Muk81] S. M...

work page arXiv 2007

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P˘ adurariu and Y

[PT23b] T. P˘ adurariu and Y. Toda,Categorical and K-theoretic Donaldson-Thomas theory of C3 (part II), Forum Math. Sigma11(2023), Paper No. e108,

2023

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P˘ adurariu and Y

[PT24a] T. P˘ adurariu and Y. Toda,Quasi-BPS categories for Higgs bundles, arXiv:2408.02168,

work page arXiv

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P˘ adurariu and Y

[PT24b] T. P˘ adurariu and Y. Toda,Categorical and K-theoretic Donaldson-Thomas theory of C3 (part I), Duke Math. J.173(2024), 1973–2038. [PT25a] T. P˘ adurariu and Y. Toda,The Dolbeault geometric Langlands conjecture via limit categories, arXiv:2508.19624,

work page arXiv 2024

[13] [13]

Topol.18 (2025), no

[PT25b] ,Topological K-theory of quasi-BPS categories for Higgs bundles, J. Topol.18 (2025), no. 4, Paper No. e70049,

2025

[14] [14]

A proof of Dolbeault geometric Langlands for $\mathrm{GL}_2$ with reduced spectral curves

THE DOLBEAULT GEOMETRIC LANGLANDS BEYOND THE ELLIPTIC LOCUS 87 [PTVV13] T. Pantev, B. To¨ en, M. Vaqui´ e, and G. Vezzosi,Shifted symplectic structures, Publ. Math. Inst. Hautes ´Etudes Sci.117(2013), 271–328. [ˇSdB17] ˇS. ˇSpenko and M. Van den Bergh,Non-commutative resolutions of quotient singularities for reductive groups, Invent. Math.210(2017), no. 1...

work page internal anchor Pith review Pith/arXiv arXiv 2013