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The derived moduli of perverse sheaves

This paper constructs higher derived Artin stacks parametrizing constructible sheaves on complex algebraic and compact real analytic varieties, demonstrating that perversity functions define open 1-Artin substacks of perverse sheaves that generalize character stacks and enable the construction of new cohomological Hall algebras for punctured Riemann surfaces.

Original authors: Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier

Published 2026-06-29

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CC BY 4.0

Original authors: Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a cartographer trying to draw a map of a very strange, jagged landscape. This landscape isn't just smooth hills and valleys; it's made of different "layers" or "strata" (like the layers of an onion, or the different materials in a geode). Some parts are smooth, some are sharp, and some are just points.

In mathematics, there are objects called sheaves. Think of a sheaf as a way of attaching a specific piece of data (like a number, a shape, or a rule) to every single point on this landscape. Usually, these rules change smoothly as you move across the smooth parts, but they can jump or behave wildly when you hit the jagged edges.

This paper is about building a master catalog (a "moduli stack") for a very special, tricky type of sheaf called a perverse sheaf.

Here is the breakdown of what the authors did, using simple analogies:

1. The Problem: The "Character" of the Map

For a long time, mathematicians have studied "character varieties." Imagine you have a group of friends (a mathematical group) and you want to see how they can interact with a specific shape (like a donut or a sphere). The "character variety" is a map showing all the possible ways these friends can interact.

However, these maps often have sharp, ugly corners (singularities) where the math breaks down. To fix this, mathematicians started using "stacks," which are like maps that keep track of not just the interactions, but also the symmetries of those interactions. This makes the map smoother and more honest.

But there was a bigger problem: The old maps only worked for "local systems" (sheaves that behave very nicely everywhere). They couldn't handle the "perverse" sheaves, which are allowed to be messy and jump around at the jagged edges of the landscape.

2. The Solution: A "Smart" Catalog

The authors (Haine, Porta, and Teyssier) built a new, high-tech catalog.

The Landscape: They looked at complex shapes (like algebraic curves) and real-world shapes (like analytic surfaces) that are chopped up into pieces (stratified spaces).
The Objects: They cataloged "constructible sheaves" (data that is consistent within each piece) and "perverse sheaves" (a specific, balanced way of handling the data at the jagged edges).
The "Derived" Twist: This is the most important part. Usually, a catalog just lists items. This new catalog is a "derived" stack.
- Analogy: Imagine a normal catalog lists "Red Car." A derived catalog doesn't just list "Red Car"; it also lists how the car is connected to other cars, the history of its paint, and the invisible forces holding it together. It captures the "shape" of the space of all possible sheaves, including all their hidden, higher-dimensional connections.

3. How They Did It: The "Exit Path"

To build this catalog without getting lost in the jagged edges, they used a concept called stratified homotopy theory.

Analogy: Imagine you are walking through a maze with different rooms (strata). A normal map just shows the rooms. An "exit path" map shows you how you can leave one room and enter another, and all the possible ways you can wiggle around inside a room before leaving.
The authors proved that for certain types of landscapes (like those found in algebraic geometry or real analytic geometry), these "exit paths" are finite and manageable. This allowed them to treat the messy landscape as a manageable, finite puzzle.

4. The Result: A Perfectly Organized Library

They proved that for these landscapes, you can build a 1-Artin stack.

Translation: This is a mathematical structure that is "locally finite" (it's not infinitely huge and unmanageable) and "locally of finite presentation" (you can describe it with a finite set of rules).
They showed that for any way you choose to define "perverse" (a specific rule for balancing the data at the edges), there is a perfect, open section in their catalog dedicated to it.

5. Why It Matters: The "Hall Algebra"

The paper ends with a cool application called a Cohomological Hall Algebra (CoHA).

Analogy: Imagine you have a collection of Lego bricks (the perverse sheaves). Usually, you just look at them. But this new structure allows you to "multiply" them. If you take two sheaves and combine them in a specific way, you get a new sheaf, and this process follows strict algebraic rules.
The authors used their new catalog to build this "multiplication machine" for sheaves on punctured Riemann surfaces (think of a donut with holes punched in it). This creates a new algebraic structure that mathematicians can use to study these shapes.

Summary

In short, the authors took a very messy, jagged mathematical problem (classifying sheaves on complex, layered shapes) and built a structured, high-dimensional catalog for them. They used the geometry of "exit paths" to tame the mess, proving that these catalogs are well-behaved and can be used to create new algebraic tools (Hall algebras) for understanding the shapes of the universe.

Technical Summary: The Derived Moduli of Perverse Sheaves

Problem Statement
The paper addresses the construction of moduli spaces for perverse sheaves on complex algebraic varieties and compact real analytic varieties. While classical character varieties (parametrizing representations of fundamental groups) and their stacky generalizations are well-understood, extending these moduli constructions to the realm of constructible and perverse sheaves presents significant challenges.

Previous approaches, such as those by Gelfand, MacPherson, and Vilonen [GMV96], rely on representing perverse sheaves as representations of a specific quiver with relations $Q(X, P)$ . This method suffers from two primary limitations:

Poor Functoriality: The quiver $Q(X, P)$ depends on auxiliary choices (e.g., projective embeddings), making it difficult to control the dependence of the moduli space on the underlying stratified space $(X, P)$ or to vary the stratification itself.
Absence of a Canonical Derived Structure: Standard methods for constructing derived enhancements of moduli spaces (via the heart of a $t$ -structure on a stable $\infty$ -category of finite type) fail here. The derived category of representations of $Q(X, P)$ is generally not of finite type (e.g., it may have infinite homological dimension), and even when it is, the resulting derived structure does not match the geometrically expected one sensitive to the stratification's cohomology.

Methodology
The authors employ a high-level synthesis of stratified homotopy theory and the theory of moduli of objects in stable $\infty$ -categories.

Stratified Homotopy Theory as a Black Box: The paper utilizes the framework of exodromic stratified spaces developed in [HPT26]. Instead of relying on quiver representations, the authors use the $\infty$ -category of exit-paths $\Pi_\infty(X, P)$ . For "nice" stratified spaces (subanalytic or algebraic), the $\infty$ -category of hyperconstructible hypersheaves is equivalent to the category of functors from $\Pi_\infty(X, P)$ to the coefficient category.
Moduli of Objects: Following Toën and Vaquié [TV07], the authors construct the moduli stack of objects for the large stable $\infty$ -category of constructible complexes, $\text{Cons}_P(X; \text{Mod}_k)$ . A key technical insight is that while this category is a priori very large, the finiteness properties of the exit-path category $\Pi_\infty(X, P)$ (proven in [HPT26]) ensure that the category of constructible complexes with perfect stalks is of "finite type" in the sense required for representability.
Perverse $t$ -Structure and Flatness: The authors define a perverse $t$ -structure on the category of constructible hypersheaves relative to a perversity function $p: P \to \mathbb{Z}$ . They establish that the condition of being "perverse" corresponds to a flatness condition relative to the base ring. Crucially, they prove that this flatness condition is an open condition (Theorem 3.3.5), allowing the extraction of the moduli of perverse sheaves as an open substack of the larger moduli of constructible complexes.

Key Contributions and Results

Construction of the Constructible Moduli Stack: The paper constructs a geometric derived stack $\text{Cons}_P(X)$ parametrizing complexes of $P$ -constructible sheaves with perfect stalks.
- Theorem 3: For a compact subanalytic or algebraic stratified space $(X, P)$ , $\text{Cons}_P(X)$ is a geometric derived stack locally of finite presentation.
- Theorem 4: The authors generalize this to allow the stratification to vary, constructing a derived stack $\text{Cons}(X)$ parametrizing sheaves constructible with respect to some subanalytic (or algebraic) stratification.
The Perverse Character Stack: By cutting out the open substack of flat families with respect to the perverse $t$ -structure, the authors obtain the moduli of perverse sheaves.
- Main Result: For any perversity function $p$ , there exists an open 1-Artin substack $\text{}_p\text{Perv}_P(X) \subset \text{Cons}_P(X)$ .
- Properties: This stack is locally of finite presentation and generalizes the usual character stacks. Unlike the quiver approach, this construction is canonical and its tangent complex is sensitive to the cohomology of the strata and their links.
Cohomological Hall Algebras (CoHA): As a demonstration of the utility of the derived structure, the authors construct a perverse cohomological Hall algebra.
- Application: For a punctured Riemann surface $X$ with marked points, the Borel–Moore homology of the derived stack of perverse sheaves admits a natural associative product (Hall convolution). This result relies on the properness of the relevant Quot schemes (established in [Lam25]) and the specific properties of the derived stack constructed in this paper.

Significance and Claims
The paper claims to provide a robust, canonical derived enhancement for the moduli of perverse sheaves that overcomes the limitations of quiver-based approaches.

Canonical Derived Structure: The derived structure is intrinsic to the geometry of the stratification. The tangent complex at a point reflects the cohomology of the strata and the links, a feature not captured by the derived category of quiver representations.
Generality: The construction drops the need for projectivity assumptions and Whitney stratifications, working instead with subanalytic and algebraic stratified spaces. It also applies to arbitrary perversity functions, not just the middle perversity.
Foundation for Future Work: The authors position this work as a foundational step. They note that the existence of a canonical shifted Poisson structure on these stacks (recently shown by Christ–Lampetti [CL26]) and the construction of good moduli spaces (Lam [Lam25]) are continuations of the results presented here. The paper explicitly states that the construction of the CoHA is a "first piece of evidence" that the derived structure is robust.

The authors emphasize that their method relies on the finiteness theorems of exit-path $\infty$ -categories from [HPT26] rather than the homological finiteness of specific quivers, allowing them to proceed in cases where the latter is unknown or fails.