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arxiv: 2606.27790 · v1 · pith:DL4QRLQ7new · submitted 2026-06-26 · 🧮 math.AG

The derived moduli of perverse sheaves

Peter J. Haine , Mauro Porta , Jean-Baptiste Teyssier This is my paper

Pith reviewed 2026-06-29 02:40 UTC · model grok-4.3

classification 🧮 math.AG
keywords derived moduli stacksperverse sheavesconstructible sheavesArtin stackscohomological Hall algebrascharacter stacksalgebraic varieties
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The pith

Higher derived Artin stacks parametrize constructible sheaves on varieties, with every perversity function cutting out an open 1-Artin substack of perverse sheaves that generalizes character stacks.

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

The paper builds higher derived Artin stacks whose points correspond to constructible sheaves on complex algebraic varieties and on compact real analytic varieties. It proves that any perversity function selects an open substack inside this moduli object. The resulting substack of perverse sheaves is itself a 1-Artin stack locally of finite presentation. A reader would care because the construction supplies a geometric home for perverse sheaves that recovers ordinary character stacks as a special case and directly produces new cohomological Hall algebras on punctured Riemann surfaces.

Core claim

We construct higher derived Artin stacks parametrizing constructible sheaves on complex algebraic varieties and compact real analytic varieties. Furthermore, we show that every perversity function gives rise to an open substack of perverse sheaves, which is a 1-Artin stack locally of finite presentation that generalizes usual character stacks. As a sample application of the derived structure, we construct new examples of cohomological Hall algebras associated to punctured Riemann surfaces.

What carries the argument

Higher derived Artin stacks for constructible sheaves, inside which perversity functions cut out open substacks.

If this is right

  • Perverse sheaves on the given varieties acquire the structure of a 1-Artin stack locally of finite presentation.
  • Usual character stacks appear as special cases inside the new moduli objects.
  • New cohomological Hall algebras can be built from the derived moduli data on punctured Riemann surfaces.
  • The same construction works uniformly for both complex algebraic varieties and compact real analytic varieties.

Where Pith is reading between the lines

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

  • The same derived-stack machinery may extend to constructible sheaves on other geometric categories once the underlying Artin-stack axioms are verified.
  • Cohomological Hall algebras arising this way could be compared directly with those obtained from quiver representations or from other sheaf-theoretic constructions.
  • The open-substack property suggests that deformation theory and obstruction theory for perverse sheaves can be read off from the derived moduli stack in a uniform way.

Load-bearing premise

The constructions of the higher derived Artin stacks for constructible sheaves can be performed in the derived setting such that the perversity condition defines an open substack without additional obstructions or failures of the Artin stack axioms in the higher derived context.

What would settle it

An explicit perversity function on a concrete variety for which the corresponding substack of perverse sheaves fails to be open, fails to be 1-Artin, or fails to be locally of finite presentation would disprove the central claim.

read the original abstract

We construct higher derived Artin stacks parametrizing constructible sheaves on complex algebraic varieties and compact real analytic varieties. Furthermore, we show that every perversity function gives rise to an open substack of perverse sheaves, which is a 1-Artin stack locally of finite presentation that generalizes usual character stacks. As a sample application of the derived structure, we construct new examples of cohomological Hall algebras associated to punctured Riemann surfaces.

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

0 major / 0 minor

Summary. The paper constructs higher derived Artin stacks parametrizing constructible sheaves on complex algebraic varieties and compact real analytic varieties. It shows that every perversity function induces an open substack of perverse sheaves that is a 1-Artin stack locally of finite presentation, generalizing usual character stacks. A sample application constructs new cohomological Hall algebras associated to punctured Riemann surfaces.

Significance. If the central claims hold, the work would extend derived algebraic geometry by furnishing derived moduli stacks for constructible and perverse sheaves, with the openness of the perverse locus and the 1-Artin property providing a direct generalization of character stacks; the Hall algebra application would supply new examples in this framework.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting the potential significance of the constructions if the central claims are verified. We are glad that the work is viewed as a natural extension of derived algebraic geometry via moduli of constructible and perverse sheaves, together with the indicated application to new cohomological Hall algebras. No specific major comments appear in the report, so we offer no point-by-point responses below. We remain available to clarify any technical points or to supply additional details should the referee wish to raise them.

Circularity Check

0 steps flagged

No circularity identified; derivation relies on external derived algebraic geometry frameworks

full rationale

The abstract presents constructions of higher derived Artin stacks for constructible sheaves and open substacks of perverse sheaves without providing equations, fitted parameters, or self-citations that reduce claims to inputs by construction. No load-bearing steps matching the enumerated circularity patterns are detectable from the given text. The work builds on prior results in derived algebraic geometry, which are treated as independent external support rather than self-referential. This is a standard non-finding for papers whose central claims remain independent of internal fits or renamings.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The work likely relies on standard axioms of derived algebraic geometry and higher category theory.

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Reference graph

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