- The paper develops a derived moduli theory constructing higher Artin stacks that parametrize perverse sheaves with derived enhancements.
- It establishes rigorous stratified homotopy foundations and functorial flatness criteria for constructible complexes on diverse geometries.
- The results yield new applications in cohomological Hall algebras and shifted Poisson structures, impacting geometric representation theory.
Derived Moduli Stacks for Perverse Sheaves: Construction and Applications
Motivation and Context
The paper "The derived moduli of perverse sheaves" (2606.27790) develops a comprehensive theory of derived moduli stacks parametrizing perverse sheaves and constructible complexes on stratified spaces, with emphasis on both complex algebraic varieties and compact real analytic varieties. The research is situated at the intersection of higher category theory, geometric representation theory, and homotopical algebra. The historical motivation arises from the study of character varieties and stacks, which parametrized representations of finitely generated groups or fundamental groups of spaces. These spaces have deep connections with low-dimensional topology, gauge theory, nonabelian Hodge theory, and the geometric Langlands program.
The classical moduli problems, such as those governed by representation theory of quivers, lacked robust functoriality and derived enhancements necessary for finer geometric and homological analysis. The authors generalize the character stack paradigm, constructing moduli stacks not only for local systems but for perverse sheaves — highly nuanced objects defined via t-structures associated to stratifications. The derived structure encodes additional homotopical information, providing precise control of singularities and enabling symplectic/Poisson geometry and Hall algebra constructions.
Main Results
Construction of Derived Artin Stacks
The central advancement is the construction of higher (and $1$-) Artin stacks parametrizing complexes of constructible sheaves with perfect stalks, and open substacks of perverse sheaves determined by a perversity function. For a stratified space (X,P) (either compact subanalytic or algebraic with finite stratification), and a connective derived commutative ring k, the authors prove:
- The existence of a geometric derived stack ConsP​(X) locally of finite presentation over k, classifying P-constructible complexes with perfect stalks;
- For every perversity function p:P→Z, there is an open $1$-Artin substack $\tensor*[^p]{\mathbf{Perv}_P}(X) \subset Cons_P(X)$ parametrizing flat families of perverse sheaves for the given stratification.
This construction is functorial in the choice of stratification and perversity, and generalizes character stacks (moduli of local systems) to the perverse and constructible field. The theory extends to variable stratifications: one can construct stacks parametrizing sheaves constructible with respect to some stratification varying in a prescribed class.
Stack-Theoretic and Homotopy-Theoretic Foundations
To enable the derived moduli construction, the paper establishes robust foundations in stratified homotopy theory, including:
- The theory of exodromic stratified spaces, which generalizes conical stratifications and ensures atomically generated $1$0-categories of constructible sheaves;
- Finiteness results for exit-path $1$1-categories of stratified spaces, extending classical Lefschetz-type theorems to the stratified, higher categorical setting;
- Functoriality and recollement structures for constructible and perverse hypersheaves, including $1$2-pullback and tensor operation compatibility.
These results ensure that the moduli constructions are geometric and locally of finite presentation whenever the underlying exit-path category exhibits categorical compactness.
Structure of Perverse Moduli Stacks and Openness
A key technical innovation is the categorical analysis of perverse $1$3-structures on the derived categories of constructible sheaves, parametrized by stratification and perversity. The authors establish precise criteria for flatness and openness: perverse flatness is equivalent to explicit Tor amplitude conditions on stalks and $1$4-restrictions. Consequently, the locus of flat families of perverse sheaves forms an open substack within the ambient derived moduli stack.
For any stratified space with suitable finiteness properties, the stack of perverse sheaves is $1$5-Artin and locally of finite presentation. The tangent complex at a point corresponding to a family $1$6 is computed as $1$7, matching expectations from derived deformation theory.
Applications: Cohomological Hall Algebras and Symplectic Structures
As a sample application, the paper constructs new examples of cohomological Hall algebras (CoHA) associated to derived moduli of perverse sheaves on punctured Riemann surfaces. For $1$8 a stratified smooth curve and suitable perversity, the Borel–Moore homology $1$9 acquires an associative multiplication via Hall convolution, and (X,P)0 becomes naturally (X,P)1-monoidal.
The paper also establishes — in future-referenced joint work — that these perverse moduli stacks admit canonical shifted Poisson structures, provided certain geometric niceness criteria are met.
Numerical and Structural Highlights
- Openness: The stack of perverse sheaves is representable by open (X,P)2-Artin substacks of the derived moduli of constructible sheaves.
- Homological Finiteness: The tangent complex of the derived moduli stack at a point (X,P)3 is (X,P)4. For perverse sheaves on curves, the internal hom is concentrated in degrees (X,P)5.
- Stack Functoriality: The construction is stable under coarsening of stratifications; moduli assignments for stratifications related by refinement form open embeddings of moduli stacks.
- Robust Derived Enhancement: The moduli stacks constructed canonically encode derived structure, sensitive to all higher homotopy data of the stratification and links.
The claims rely on sophisticated categorical machinery, including recent finiteness and coarsening theorems for exit-path (X,P)6-categories.
Implications and Future Directions
The framework for derived moduli of perverse sheaves creates powerful new tools for geometric representation theory, expanding the scope of moduli spaces available for quantization, Hall algebra constructions, and Poisson/symplectic geometry. The shifting of focus from objects in abelian categories (representations of quivers) to perverse sheaves with derived enhancements allows refined stratification-dependent analysis, potentially impacting aspects of the geometric Langlands program and related areas.
The functoriality and geometricity developed here are pivotal for derived stack theory and categorical approaches to moduli problems. Future work is indicated on shifted Poisson structures and further applications to Lagrangian geometry, as well as on universal moduli constructions over variable stratifications.
Conclusion
This paper rigorously establishes the existence and geometric properties of derived moduli stacks for perverse sheaves and constructible complexes on stratified spaces, integrating higher categorical and homotopical methods. It provides a canonical derived enhancement, resolves significant issues in functoriality and singularity control, and enables new applications in cohomological Hall algebra theory. The methods and results position the derived approach to perverse sheaf moduli as a central tool for both theoretical investigations and future developments in algebraic geometry, topology, and mathematical physics.