Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem

Efe \.Izbudak; Kadri \.Ilker Berktav

arxiv: 2605.08394 · v2 · pith:DKDAATXCnew · submitted 2026-05-08 · 🧮 math.AG

Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem

Efe \.Izbudak , Kadri \.Ilker Berktav This is my paper

Pith reviewed 2026-06-30 22:49 UTC · model grok-4.3

classification 🧮 math.AG

keywords derived symplectic geometrycontact structuresLegendrian intersectionsG_m-actionsderived algebraic geometrymoduli stacksjet bundlesHiggs bundles

0 comments

The pith

Quotients by weight-1 G_m-actions turn derived symplectic spaces into contact structures on hypersurfaces.

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

The paper translates the classical transversality lemma of contact geometry into derived algebraic geometry by replacing the Liouville vector field with the fundamental vector field of a weight-1 G_m-action. It establishes that the quotient of a derived symplectic space by this action descends the symplectic data to a contact structure while avoiding a transverse hypersurface. The argument then uses base change, an infinity-categorical descent cube, and G_m-equivariant lifts along the symplectification projection to prove a derived Legendrian intersection theorem. These tools are applied to show that discriminant loci of 1-jet bundles carry a (-1)-shifted contact structure and to produce contact structures on certain derived moduli stacks.

Core claim

The central claim is that taking the quotient of a derived symplectic space descends the symplectic data to a contact structure on a hypersurface transverse to the fundamental vector field of a weight-1 G_m-action, which replaces the classical Liouville vector field in the derived setting; the derived Legendrian intersection theorem then follows from base change, an infinity-categorical descent cube, and G_m-equivariant lifts along the symplectification projection.

What carries the argument

The equivariant quotient of a derived symplectic space by a weight-1 G_m-action, which carries symplectic data to a contact structure via the fundamental vector field.

If this is right

Discriminant loci of 1-jet bundles carry a (-1)-shifted contact structure.
The construction applies to moduli of projective Higgs bundles, l-adic local systems, and Lie 2-groups.
Additional examples of contact derived moduli stacks arise from the same quotient procedure.

Where Pith is reading between the lines

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

The method may supply contact structures on further derived moduli spaces whose underlying objects admit natural G_m-actions.
The derived Legendrian intersection theorem could yield new intersection formulas when applied to stacks of local systems or Higgs bundles.
Extensions to other reductive group actions might produce analogous descents from higher shifted symplectic data.

Load-bearing premise

G_m-equivariant lifts along the symplectification projection exist and permit base change together with an infinity-categorical descent cube.

What would settle it

A derived symplectic space equipped with a weight-1 G_m-action for which no equivariant lift exists along the symplectification projection, or for which the descent cube fails to produce a contact structure on the quotient hypersurface.

read the original abstract

The classical transversality lemma of contact geometry constructs a contact structure on a hypersurface transverse to a Liouville vector field using point-set topology and local flows. This paper translates the classical transversality lemma into the context of derived algebraic geometry and provides the derived Legendrian intersection theorem, along with various applications to moduli theory. In brief, we first prove that taking the quotient of a derived symplectic space descends the symplectic data to a contact structure, avoiding a transverse hypersurface, where the fundamental vector field of a weight 1 $\mathbb{G}_m$-action, in the derived setting, replaces the classical Liouville vector field. Secondly, the derived Legendrian intersection theorem is proven using base change, an $\infty$-categorical descent cube, and $\mathbb{G}_m$-equivariant lifts along the symplectification projection. As applications of the main results, we first examine the derived geometry of the discriminant loci of 1-jet bundles and show that these loci carry a $(-1)$-shifted contact structure. In addition, we show that our results apply to certain moduli problems, including projective Higgs bundles, $\ell$-adic local systems, and Lie 2-groups, and we provide further examples of contact derived moduli stacks.

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

The paper moves the classical transversality lemma into derived geometry via G_m-quotients on symplectic spaces to produce contact structures and a Legendrian intersection theorem, but the required equivariant lifts are asserted rather than generally secured.

read the letter

The core move is to replace the classical Liouville vector field with the fundamental vector field of a weight-1 G_m-action and take the quotient of a derived symplectic space to descend to a contact structure without a transverse hypersurface. From there the authors build a derived Legendrian intersection theorem by base change plus an infinity-categorical descent cube that needs G_m-equivariant lifts along the symplectification projection. The applications then check that the discriminant loci of 1-jet bundles carry a (-1)-shifted contact structure and that the same construction works for moduli of projective Higgs bundles, l-adic local systems, and Lie 2-groups.

The applications are the clearest part. They give explicit derived moduli stacks where the contact data appears, which is the sort of output that people working on moduli problems can actually use. The translation of the classical lemma is also stated cleanly.

The weakest step is the existence and compatibility of the G_m-equivariant lifts. The abstract says these lifts “permit” the descent cube, yet supplies no general criterion or obstruction theory that would guarantee the lifts on an arbitrary derived symplectic stack. If the lifts fail to exist or fail to make the cube a pullback, both the contact descent and the intersection theorem are blocked. The stress-test note correctly flags this as the load-bearing assumption that is least secured in the given description. The rest of the argument uses standard base-change and descent tools, so the issue is localized.

This paper is for people already working in derived symplectic and contact geometry who want to see the classical statements lifted to stacks and applied to concrete moduli examples. A reader who knows the classical transversality lemma and the relevant infinity-categorical machinery will follow the claims. It is worth sending to peer review so the lift construction can be checked in detail; the applications alone make the work worth referee time even if the general statement needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to translate the classical transversality lemma of contact geometry into derived algebraic geometry. It proves that the quotient of a derived symplectic space by a weight-1 G_m-action descends the symplectic data to a contact structure, with the fundamental vector field replacing the Liouville vector field. It then establishes the derived Legendrian intersection theorem via base change, an ∞-categorical descent cube, and G_m-equivariant lifts along the symplectification projection. Applications are provided to the derived geometry of discriminant loci of 1-jet bundles carrying (-1)-shifted contact structures, and to moduli problems including projective Higgs bundles, ℓ-adic local systems, and Lie 2-groups.

Significance. If the results hold, this work would supply a derived-algebraic replacement for the classical Liouville vector field and a Legendrian intersection theorem that applies directly to moduli stacks. The applications to discriminant loci and several moduli problems indicate that the framework could be used to equip additional derived moduli spaces with contact structures without constructing transverse hypersurfaces.

major comments (2)

[Proof of the derived Legendrian intersection theorem] The proof of the derived Legendrian intersection theorem (second paragraph of the abstract and the corresponding section) relies on the existence of G_m-equivariant lifts along the symplectification projection so that the ∞-categorical descent cube is a pullback. No general existence criterion or obstruction theory is supplied for arbitrary derived symplectic stacks; the argument only verifies the lifts on the listed moduli examples. This assumption is load-bearing for both the contact descent and the intersection theorem.
[Quotient construction and descent of symplectic data] In the construction that quotients a derived symplectic space by the weight-1 G_m-action to obtain the contact structure, the non-degeneracy of the descended 2-form (or its derived analogue) after quotienting is asserted but not shown to follow from the classical Liouville condition in a way that is independent of the specific examples.

minor comments (2)

The abstract and introduction would benefit from an explicit statement of the precise ∞-categorical setting (e.g., which model of derived stacks is used) before the descent-cube argument is invoked.
Notation for shifts and for the fundamental vector field of the G_m-action should be introduced once and used consistently in all statements of the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments. We address the two major comments point by point below, indicating planned revisions where the manuscript requires clarification or strengthening.

read point-by-point responses

Referee: [Proof of the derived Legendrian intersection theorem] The proof of the derived Legendrian intersection theorem (second paragraph of the abstract and the corresponding section) relies on the existence of G_m-equivariant lifts along the symplectification projection so that the ∞-categorical descent cube is a pullback. No general existence criterion or obstruction theory is supplied for arbitrary derived symplectic stacks; the argument only verifies the lifts on the listed moduli examples. This assumption is load-bearing for both the contact descent and the intersection theorem.

Authors: We agree that the existence of G_m-equivariant lifts along the symplectification projection is a key hypothesis required for the ∞-categorical descent cube to be a pullback, and thus for the derived Legendrian intersection theorem. The manuscript verifies the existence of these lifts only in the listed moduli examples (projective Higgs bundles, ℓ-adic local systems, Lie 2-groups) rather than supplying a general existence criterion or obstruction theory for arbitrary derived symplectic stacks. This is a genuine limitation of the present work; we do not claim the result holds without this assumption. We will revise the statement of the theorem (both in the abstract and the main section) to make the assumption explicit, and we will add a remark discussing its scope and the fact that it has been checked only in the applications. revision: partial
Referee: [Quotient construction and descent of symplectic data] In the construction that quotients a derived symplectic space by the weight-1 G_m-action to obtain the contact structure, the non-degeneracy of the descended 2-form (or its derived analogue) after quotienting is asserted but not shown to follow from the classical Liouville condition in a way that is independent of the specific examples.

Authors: We acknowledge that the non-degeneracy of the descended 2-form after quotienting by the weight-1 G_m-action needs to be derived explicitly from the Liouville-type condition on the original derived symplectic form, rather than being asserted in a manner that appears to rely on the examples. The manuscript intends the argument to be general, but we agree that the current exposition does not make the independence from specific examples sufficiently clear. We will revise the relevant section to include a self-contained argument showing that non-degeneracy of the descended form follows directly from the G_m-action and the original symplectic data in the derived setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on standard base change and descent without reduction to inputs by construction.

full rationale

The paper's derivation proceeds by defining the contact structure via G_m-quotient of a derived symplectic space (replacing Liouville field with fundamental vector field) and then invoking base change plus an ∞-categorical descent cube that requires G_m-equivariant lifts along symplectification. These steps are presented as applications of existing ∞-categorical machinery rather than self-definitions, fitted predictions, or load-bearing self-citations. No equations equate the output contact data or intersection theorem to the input symplectic data by construction, and no uniqueness theorems or ansatzes are smuggled via prior self-citations. The existence of the lifts is an explicit enabling assumption, not a hidden tautology. The derivation is therefore self-contained against external categorical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background assumptions in derived algebraic geometry and contact/symplectic geometry; no free parameters, new invented entities, or ad-hoc axioms are apparent from the abstract description.

axioms (2)

standard math Standard axioms and constructions of derived algebraic geometry, including shifted symplectic structures and G_m-actions
The paper operates within this established framework as described in the abstract.
domain assumption Existence of weight-1 G_m-actions on derived symplectic spaces whose fundamental vector fields can replace Liouville fields
Invoked to construct the contact structure via quotient without transverse hypersurface.

pith-pipeline@v0.9.1-grok · 5760 in / 1433 out tokens · 32419 ms · 2026-06-30T22:49:10.783372+00:00 · methodology

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Forward citations

Cited by 3 Pith papers

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Constructs the derived Legendrian category F_c(X) for n-shifted contact derived Artin stacks and the (∞,2)-category Leg_n of Legendrian correspondences, with applications to moduli theory.

Reference graph

Works this paper leans on

26 extracted references · 7 canonical work pages · cited by 2 Pith papers · 2 internal anchors

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Lurie,Higher Topos Theory, Ann

J. Lurie,Higher Topos Theory, Ann. Math. Stud., 170. Princeton University Press, Princeton, NJ, 2009

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Lurie,Higher Algebra, Unpublished manuscript, available at author’s website, 2017

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B. To ¨en, G. Vezzosi,Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224 pp

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Pantev, B

T. Pantev, B. To ¨en, M. Vaqui ´e, and G. Vezzosi,Shifted symplectic structures, Publications math ´ematiques de l’IH´ES, 117:271–328, 2013

2013

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Calaque, T

D. Calaque, T. Pantev, B. To¨en, M. Vaqui´e, and G. Vezzosi,Shifted Poisson Structures and deformation quantiza- tion, Journal of Topology, 10(2):483–584, 2017

2017

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Calaque,Shifted cotangent stacks are shifted symplectic, Annales de la Facult ´e des sciences de Toulouse: Math´ematiques, 28(1):67–90, 2019

D. Calaque,Shifted cotangent stacks are shifted symplectic, Annales de la Facult ´e des sciences de Toulouse: Math´ematiques, 28(1):67–90, 2019

2019

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K. ˙I. Berktav,Shifted Contact Structures and Their Local Theory, Ann. Fac. Sci. Toulouse, Math., Serie 6, Vol. 33(4): 1019-1057, 2024

2024

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K. ˙I. Berktav,On shifted contact derived Artin stacks, Higher Structures 9(2):103–135, 2025

2025

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K. ˙I. Berktav,Legendrian Structures in Derived Geometry, 2025

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To ¨en,Derived Algebraic Geometry, EMS Surveys in Mathematical Sciences, 1(2):153–240, 2014

B. To ¨en,Derived Algebraic Geometry, EMS Surveys in Mathematical Sciences, 1(2):153–240, 2014

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Safronov

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work page arXiv 2023

[13] [13]

Iwanari,Tannakization in derived algebraic geometry, Journal of K-theory, 14(3) 642–700, 2014, doi:10.1017/is014008019jkt278

I. Iwanari,Tannakization in derived algebraic geometry, Journal of K-theory, 14(3) 642–700, 2014, doi:10.1017/is014008019jkt278

work page doi:10.1017/is014008019jkt278 2014

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From HAG to DAG: derived moduli stacks. In Axiomatic, enriched and motivic homotopy theory

To ¨en, B. Vezzosi, G. 2004. “From HAG to DAG: derived moduli stacks. In Axiomatic, enriched and motivic homotopy theory”, Springer Netherlands, 173-216

2004

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1979

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1987

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R. T ´ellez Dom ´ınguez,Moduli of Higher Connections and Holomorphic 2-Bundles, PhD Thesis, Universidad Aut´onoma de Madrid, 2025

2025

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J. S. Milne, ´Etale Cohomology, Princeton Mathematical Series, 33. Princeton University Press, 1980

1980

[19] [19]

Moduli of $\ell$-adic pro-\'etale local systems for smooth non-proper schemes

J. Ant ´onio,Moduli ofℓ-adic pro-´ etale local systems for smooth non-proper schemes, arXiv:1904.08001, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1904

[20] [20]

J. P . Pridham,Deformingℓ-adic representations of the fundamental group of a smooth variety, Journal of Algebraic Geometry, 15:415–442, 2006

2006

[21] [21]

Arinkin, D

D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin, N. Rozenblyum, Y. Varshavsky,The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support, arXiv:2010.01906, 2020

work page arXiv 2010

[22] [22]

Singular support of coherent sheaves, and the geometric Langlands conjecture

D. Arinkin, D. Gaitsgory,Singular support of coherent sheaves and the geometric Langlands conjecture, arXiv:1201.6343, 2012

work page internal anchor Pith review Pith/arXiv arXiv 2012

[23] [23]

Zhu,Coherent sheaves on the stack of Langlands parameters, Proceedings of Symposia in Pure Mathematics, 2020

X. Zhu,Coherent sheaves on the stack of Langlands parameters, Proceedings of Symposia in Pure Mathematics, 2020

2020

[24] [24]

Jiang,On the construction of moduli stack of projective Higgs bundles over surfaces, arXiv:1911.00250, 2019

Y. Jiang,On the construction of moduli stack of projective Higgs bundles over surfaces, arXiv:1911.00250, 2019

work page arXiv 1911

[25] [25]

Jiang, M

Y. Jiang, M. Kool,Twisted sheaves and SU(r)/Z r Vafa-Witten theory, arXiv:2006.10368, 2020

work page arXiv 2006

[26] [26]

Ben-Bassat, S

O. Ben-Bassat, S. Das, T. Pantev,Moduli stacks of Higgs bundles on stable curves, Moduli 3, e7, 1-44, 2026

2026