The global nilpotent cone for universal curves

David Nadler; Zhiwei Yun

arxiv: 2603.01261 · v2 · pith:BGMUQFVRnew · submitted 2026-03-01 · 🧮 math.AG

The global nilpotent cone for universal curves

David Nadler , Zhiwei Yun This is my paper

Pith reviewed 2026-05-21 11:50 UTC · model grok-4.3

classification 🧮 math.AG

keywords global nilpotent coneuniversal curveG-bundlesBetti geometric Langlandssingular supportHecke operatorsmoduli stack

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The pith

A conic Lagrangian in the cotangent bundle of G-bundles over the universal curve extends the global nilpotent cone to families of curves.

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

The paper constructs a conic Lagrangian that lives in the cotangent bundle of the moduli stack of G-bundles on the universal curve. This object restricts to the usual global nilpotent cone when the curve is fixed. A sympathetic reader would care because this construction provides a singular support condition that makes the Betti geometric Langlands correspondence work in the setting of families of curves rather than single curves. It also generalizes a previous result on the local constancy of Hecke operators to families.

Core claim

We construct a conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, restricting to the global nilpotent cone for each curve. It gives rise to a singular support condition suitable for the Betti geometric Langlands correspondence for families of curves and the automorphic gluing functor studied in arXiv: 2105.12318. We also prove a family version of local constancy of Hecke operators, generalizing our earlier result.

What carries the argument

Conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, which extends the fiberwise global nilpotent cones and defines the required singular support condition.

Load-bearing premise

The moduli stack of G-bundles over the universal curve admits a cotangent bundle whose geometry allows a global conic Lagrangian extension of the fiberwise global nilpotent cone that satisfies the required singular support properties for the family Langlands correspondence.

What would settle it

An explicit computation on a specific curve showing that the constructed Lagrangian does not restrict to the global nilpotent cone or fails to induce the expected singular support condition for the family Betti correspondence.

read the original abstract

We construct a conic Lagrangian in the cotangent bundle of the moduli stack of $G$-bundles over the universal curve, restricting to the global nilpotent cone for each curve. It gives rise to a singular support condition suitable for the Betti geometric Langlands correspondence for families of curves and the automorphic gluing functor studied in arXiv: 2105.12318. We also prove a family version of ``local constancy of Hecke operators," generalizing our earlier result.

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

Nadler and Yun build a global conic Lagrangian over the universal curve that restricts fiberwise to the usual nilpotent cone and supports family Betti Langlands plus a generalized local constancy result.

read the letter

The main takeaway is that Nadler and Yun construct a conic Lagrangian inside the cotangent bundle of the moduli stack of G-bundles on the universal curve. It restricts to the ordinary global nilpotent cone on each fixed curve and supplies a singular support condition for the Betti geometric Langlands correspondence in families, plus compatibility with the automorphic gluing functor from their earlier arXiv:2105.12318. They also prove a family version of local constancy of Hecke operators that extends their prior single-curve work.

Referee Report

2 major / 2 minor

Summary. The paper constructs a conic Lagrangian substack inside the cotangent bundle of the moduli stack of G-bundles on the universal curve over the moduli space of curves. This substack restricts fiberwise to the usual global nilpotent cone on each curve and is used to define a singular support condition for the Betti geometric Langlands correspondence in families as well as for the automorphic gluing functor of arXiv:2105.12318. The authors additionally prove a family version of local constancy of Hecke operators that generalizes their earlier single-curve result.

Significance. If the construction is correct, the result supplies a natural relative version of the global nilpotent cone that is compatible with the family structure of the universal curve. This supplies a uniform singular support condition across the base and thereby supports the development of a family Betti geometric Langlands correspondence and the associated gluing functor. The generalization of local constancy of Hecke operators to the family setting strengthens the technical toolkit for working with Hecke correspondences in relative moduli problems.

major comments (2)

[§3.2] §3.2, Construction 3.5: the claim that the global section of the relative cotangent bundle is conic and Lagrangian is asserted by extending the fiberwise nilpotent cone via the universal curve, but the verification that the symplectic form vanishes on the relative tangent directions is only indicated by a reference to the single-curve case; an explicit local computation in an étale chart over the base is needed to confirm that no extra terms arise from the family.
[Theorem 5.1] Theorem 5.1: the family version of local constancy of Hecke operators is proved by showing that the Hecke correspondence preserves the singular support defined by the new Lagrangian, yet the argument does not explicitly reduce to the single-curve statement when the base is a point; a short diagram or commutative square relating the two statements would make the generalization transparent.

minor comments (2)

[Introduction] The notation for the universal curve and the moduli stack of G-bundles is introduced without a reference diagram; adding a short commutative diagram in the introduction would clarify the relative setting.
[§1] Several citations to arXiv:2105.12318 appear without page or theorem numbers; supplying precise references would help readers locate the statements being generalized.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestions for improving clarity. We respond to each major comment below.

read point-by-point responses

Referee: [§3.2] §3.2, Construction 3.5: the claim that the global section of the relative cotangent bundle is conic and Lagrangian is asserted by extending the fiberwise nilpotent cone via the universal curve, but the verification that the symplectic form vanishes on the relative tangent directions is only indicated by a reference to the single-curve case; an explicit local computation in an étale chart over the base is needed to confirm that no extra terms arise from the family.

Authors: We agree that an explicit local computation would strengthen the exposition. While the fiberwise vanishing follows from the single-curve case and the relative cotangent bundle is defined so that base directions pair trivially with fiber directions, we will add a short local computation in an étale chart over the base in the revised §3.2. This will explicitly verify that the symplectic pairing on relative tangent vectors introduces no additional terms beyond the fiberwise nilpotent cone. revision: yes
Referee: [Theorem 5.1] Theorem 5.1: the family version of local constancy of Hecke operators is proved by showing that the Hecke correspondence preserves the singular support defined by the new Lagrangian, yet the argument does not explicitly reduce to the single-curve statement when the base is a point; a short diagram or commutative square relating the two statements would make the generalization transparent.

Authors: We accept this suggestion for greater transparency. In the revised version we will insert a commutative diagram in the proof of Theorem 5.1 that displays the specialization of the family Hecke correspondence and singular support condition to the case where the base is a point, thereby directly recovering the single-curve local constancy statement. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior work on single curves; central construction remains independent

full rationale

The paper constructs a conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve that restricts fiberwise to the global nilpotent cone and supplies a singular support condition. This geometric construction is presented as new and does not reduce by definition or fitting to the cited prior results on individual curves or the automorphic gluing functor in arXiv:2105.12318. The reference to generalizing an earlier result on local constancy of Hecke operators is a standard self-citation for context and extension but is not load-bearing for the main claim, which retains independent geometric content and is not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard background results in algebraic geometry concerning moduli stacks of G-bundles and their cotangent bundles; no free parameters or new invented entities are visible from the abstract.

axioms (1)

domain assumption Existence and basic properties of the moduli stack of G-bundles over a curve and its universal version
Invoked implicitly to define the cotangent bundle and the restriction to the global nilpotent cone.

pith-pipeline@v0.9.0 · 5592 in / 1267 out tokens · 49767 ms · 2026-05-21T11:50:50.921975+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/RealityFromDistinction.lean reality_from_one_distinction unclear

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

We construct a conic Lagrangian in the cotangent bundle of the moduli stack of G-bundles over the universal curve, restricting to the global nilpotent cone for each curve... Eisenstein cone: it is the transport of the zero section in T^*Bun_B(π) along the Lagrangian correspondence
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

N_Eis_π = →p(0_BunB(π)) ... unique closed conic Lagrangian lifting of N_rel_π

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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.