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arxiv: 2606.11179 · v1 · pith:35CEDRALnew · submitted 2026-06-09 · 🧮 math.AG

Equivariant Contact Darboux Quotients and Perversely Categorified Legendrian Correspondences

Efe \.Izbudak This is my paper

Pith reviewed 2026-06-27 11:15 UTC · model grok-4.3

classification 🧮 math.AG
keywords equivariant Darboux theoremshifted contact stacksperverse sheavesLegendrian correspondencesderived Artin stacksenumerative invariantsFourier-Mukai functorsmicrolocal sheaf theory
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The pith

Equivariant Darboux quotients give local models for -1-shifted contact derived Artin stacks and support perverse sheaves with monodromy for enumerative invariants.

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

The paper proves an equivariant Darboux theorem for -1-shifted contact derived Artin stacks, showing they admit smooth atlases by the derived contact Darboux scheme and its quotients under reductive stabilizers. This local model allows construction of an l-adic perverse sheaf on oriented stacks, equipped with tame geometric monodromy via Verdier specialization. The sheaf extracts derived enumerative invariants through the l-adic Grothendieck-Lefschetz trace, addressing generic topological acyclicity. The work also formulates a contact analogue of Joyce's conjecture and builds categorified Legendrian 2-categories via l-adic Fourier-Mukai pull-push functors.

Core claim

We establish an equivariant Darboux theorem for -1-shifted contact derived Artin stacks. In the smooth topology these stacks admit smooth atlases by the derived contact Darboux scheme Δloc(s) and, with reductive stabilizers G, the equivariant geometric quotient [Δloc(s)/G]. By applying the BBDJS minimal model to the derived symplectification and descending algebraically along the structural free Gm-action, we construct an l-adic perverse sheaf on any oriented -1-shifted contact stack equipped with tame geometric monodromy automorphism T. This structure allows extraction of derived enumerative invariants via the l-adic Grothendieck-Lefschetz trace. The content of the other main results relies

What carries the argument

The equivariant geometric quotient stack [Δloc(s)/G] of the derived contact Darboux scheme, which provides the local atlas and supports descent of the perverse sheaf with monodromy.

If this is right

  • Derived intersections of n-shifted Legendrians produce (n-1)-shifted contact stacks that inherit the equivariant Darboux atlases.
  • The non-linear 2-categories of Legendrians admit linearizations as categorified 2-categories L Fc(X) and LLeg_0 via l-adic Fourier-Mukai pull-push functors.
  • A contact analogue of Joyce's conjecture can be formulated to linearize the categorical structures of Legendrians.
  • Derived enumerative invariants become extractable on oriented -1-shifted contact stacks even when generic topological acyclicity holds.

Where Pith is reading between the lines

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

  • The construction may extend to produce similar perverse sheaves on higher-shifted contact or symplectic stacks once Darboux models are available.
  • The tame monodromy T could interact with other monodromy actions in microlocal sheaf theory to produce refined invariants.
  • The link between derived contact moduli spaces and microlocal sheaves suggests that Legendrian correspondences might admit descriptions in terms of Fourier-Mukai transforms on the base stacks.

Load-bearing premise

The results rest on the prior establishment of smooth Darboux atlases for shifted contact derived Artin stacks together with the applicability of the BBDJS minimal model and Verdier's specialization equivalence for monodromic sheaves.

What would settle it

A concrete -1-shifted contact derived Artin stack with reductive stabilizer where the derived contact Darboux scheme fails to give a smooth atlas or where no tame geometric monodromy automorphism T can be defined on the associated perverse sheaf.

read the original abstract

Prior work has shown that shifted contact derived Artin stacks admit smooth Darboux atlases. However, establishing enumerative invariants and linearizing these categorical structures requires equivariant local models. We establish an equivariant Darboux theorem for $-1$-shifted contact derived Artin stacks. We prove that, in the smooth topology, these stacks admit smooth atlases by the derived contact Darboux scheme $\Delta\mathrm{loc}(s)$ associated to the derived discriminant locus of a relative section $s$. In the presence of reductive stabilizers $G$, this refines to the equivariant geometric quotient stack $[\Delta\mathrm{loc}(s)/G]$. By applying the BBDJS minimal model to the derived symplectification and descending algebraically along the structural free $\mathbb{G}_m$-action, we construct an $\ell$-adic perverse sheaf on any oriented $-1$-shifted contact stack. We utilize Verdier's specialization equivalence for monodromic sheaves to equip this perverse sheaf with a tame geometric monodromy automorphism $T$. This structure allows for the extraction of derived enumerative invariants via the $\ell$-adic Grothendieck-Lefschetz trace, thereby resolving the issue of generic topological acyclicity. The content of the other main results in this paper relies on a prior work, in which we have shown that derived intersections of $n$-shifted Legendrians yield $(n-1)$-shifted contact stacks and formulated the non-linear 2-categories of Legendrians $\mathcal{F}_c(X)$ and $Leg_n$. Using this geometric setup, we formulate in this paper a contact analogue of Joyce's conjecture to linearize these structures. We then construct the categorified Legendrian 2-categories $\mathfrak{L}\mathcal{F}c(X)$ and $LLeg_0$ via $\ell$-adic Fourier-Mukai pull-push functors, connecting the study of derived contact moduli spaces to microlocal sheaf theory.

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

3 major / 2 minor

Summary. The paper establishes an equivariant Darboux theorem for -1-shifted contact derived Artin stacks: in the smooth topology these admit atlases by the derived contact Darboux scheme Δloc(s) associated to the discriminant locus of a relative section s, refining to the geometric quotient stack [Δloc(s)/G] when stabilizers are reductive. Using the BBDJS minimal model on the derived symplectification and descending along the free Gm-action, together with Verdier specialization for monodromic sheaves, it constructs an ℓ-adic perverse sheaf equipped with tame geometric monodromy T on any oriented -1-shifted contact stack; the Grothendieck-Lefschetz trace then yields derived enumerative invariants. Building on prior work showing that derived intersections of n-shifted Legendrians produce (n-1)-shifted contact stacks and defining the non-linear 2-categories Fc(X) and Legn, the paper formulates a contact analogue of Joyce's conjecture and constructs the categorified 2-categories L Fc(X) and LLeg0 via ℓ-adic Fourier-Mukai pull-push functors.

Significance. If the technical steps hold, the equivariant local models and perverse-sheaf construction would supply a concrete route to linearizing Legendrian 2-categories and extracting enumerative invariants from derived contact moduli spaces, linking derived algebraic geometry with microlocal sheaf theory. The explicit use of BBDJS minimal models and Verdier equivalence in the derived-stack setting, together with the formulation of the contact Joyce conjecture, would constitute a substantive advance provided the applicability of the cited tools is verified.

major comments (3)
  1. [Abstract] Abstract (paragraph on main constructions): the construction of the ℓ-adic perverse sheaf with tame monodromy T rests on applying the BBDJS minimal model to the derived symplectification and invoking Verdier's specialization equivalence; the manuscript must supply a precise verification that these tools extend to the derived Artin stack setting with the given Gm-action, as this step is load-bearing for the enumerative-invariant claim.
  2. [Abstract] Abstract (final paragraph): the categorified 2-categories L Fc(X) and LLeg0 are defined via pull-push functors that rely on the prior work establishing that derived Legendrian intersections yield (n-1)-shifted contact stacks; an explicit statement of which results from the prior paper are invoked and where new checks are performed is required to confirm the constructions are not circular.
  3. [Abstract] Abstract (equivariant Darboux statement): the refinement to the equivariant geometric quotient [Δloc(s)/G] for reductive G is asserted without an indicated reference to the precise stack-theoretic quotient construction or the required reductivity hypothesis; this needs a dedicated paragraph or lemma showing that the atlas descends under the group action.
minor comments (2)
  1. [Abstract] Notation: the symbol Δloc(s) is introduced without an immediate definition or reference to its construction from the discriminant locus; a short clarifying sentence would improve readability.
  2. [Abstract] The phrase "smooth topology" is used without definition in the derived Artin stack context; a brief parenthetical or footnote would clarify its meaning.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments identify areas where additional explicit verification and clarification will strengthen the manuscript. We address each major comment below and will incorporate the suggested revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on main constructions): the construction of the ℓ-adic perverse sheaf with tame monodromy T rests on applying the BBDJS minimal model to the derived symplectification and invoking Verdier's specialization equivalence; the manuscript must supply a precise verification that these tools extend to the derived Artin stack setting with the given Gm-action, as this step is load-bearing for the enumerative-invariant claim.

    Authors: We agree that a self-contained verification is necessary for the load-bearing step. The current text invokes the BBDJS minimal model and Verdier specialization by reference to their established use in derived algebraic geometry, but does not spell out the checks for the free Gm-action on the symplectification in the Artin-stack context. In the revised manuscript we will insert a new lemma (placed in Section 3) that verifies the hypotheses of both tools hold under the given Gm-action, including the required tameness and monodromy conditions on the derived stack. revision: yes

  2. Referee: [Abstract] Abstract (final paragraph): the categorified 2-categories L Fc(X) and LLeg0 are defined via pull-push functors that rely on the prior work establishing that derived Legendrian intersections yield (n-1)-shifted contact stacks; an explicit statement of which results from the prior paper are invoked and where new checks are performed is required to confirm the constructions are not circular.

    Authors: We concur that an explicit accounting of dependence on the prior paper is required. The constructions invoke Theorems 3.2 and 4.1 of the earlier work to obtain the (n-1)-shifted contact stacks, together with the definition of the non-linear 2-categories Fc(X) and Legn. The novel verifications performed here concern the preservation of the perverse t-structure by the ℓ-adic Fourier-Mukai functors in the equivariant setting; these appear in Section 5. The revision will add a short clarifying paragraph at the start of Section 5 that lists the precise prior results used and isolates the new checks. revision: yes

  3. Referee: [Abstract] Abstract (equivariant Darboux statement): the refinement to the equivariant geometric quotient [Δloc(s)/G] for reductive G is asserted without an indicated reference to the precise stack-theoretic quotient construction or the required reductivity hypothesis; this needs a dedicated paragraph or lemma showing that the atlas descends under the group action.

    Authors: The referee is correct that the descent step requires a dedicated justification. The reductivity hypothesis on G guarantees that the quotient is an algebraic stack in the smooth topology; we will add a new lemma in Section 2.3 that (i) recalls the standard stack-theoretic construction of the geometric quotient [X/G] for reductive G, (ii) verifies that the contact atlas Δloc(s) is G-equivariant, and (iii) shows that the atlas descends to the quotient stack. The lemma will include the necessary reference to the relevant stack-quotient results. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper explicitly builds new results (equivariant Darboux theorem for -1-shifted contact stacks, perverse sheaf via BBDJS minimal model and Verdier specialization, and categorified Legendrian 2-categories via Fourier-Mukai functors) on top of cited prior work by the same author concerning Legendrian intersections and the base 2-categories F_c(X) and Leg_n. However, no derivation step within this manuscript reduces by construction or equation to its own inputs; the self-citations supply external foundational geometry while the current claims add independent content (equivariant quotients, monodromy automorphism T, Grothendieck-Lefschetz extraction). The derivation chain remains self-contained against external benchmarks and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper rests on domain assumptions from the author's prior results about Darboux atlases and Legendrian intersections, plus standard tools from derived geometry and sheaf theory whose applicability here is not independently verified in the abstract.

axioms (4)
  • domain assumption Shifted contact derived Artin stacks admit smooth Darboux atlases
    Invoked as established in prior work by the author.
  • domain assumption Derived intersections of n-shifted Legendrians yield (n-1)-shifted contact stacks
    Stated as shown in the author's prior work.
  • domain assumption BBDJS minimal model applies to the derived symplectification
    Used to descend algebraically along the Gm-action.
  • domain assumption Verdier's specialization equivalence holds for the monodromic sheaves in this context
    Used to equip the perverse sheaf with tame geometric monodromy T.

pith-pipeline@v0.9.1-grok · 5896 in / 1773 out tokens · 28366 ms · 2026-06-27T11:15:48.219475+00:00 · methodology

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