Perverse Sheaves: t-Structures and Applications

• Perverse sheaves are complexes in constructible derived categories characterized by a perverse t-structure that controls both support and cosupport conditions.
• They extend across algebraic, analytic, and ℓ-adic frameworks, linking intersection cohomology with representation theory through explicit D-module correspondences.
• Their combinatorial models, gluing techniques, and tensor structures enable practical applications in singularity theory, categorical wall-crossing, and mathematical physics.

Perverse sheaves are a distinguished class of complexes in the derived category of constructible sheaves on a topological space or scheme, characterized by a delicate balance of support and co-support conditions determined by a chosen stratification and a perversity function. The theory underpins modern intersection cohomology and plays a critical role across algebraic geometry, representation theory, singularity theory, and mathematical physics. Perverse sheaves are defined as the heart of a specific t-structure (the perverse t-structure) in the derived category, and their abelian category exhibits finite length and rich duality properties. The construct encompasses extensions to various settings, including analytic, étale, and ℓ-adic sheaf categories, as well as deep connections to D-modules, Tannakian formalism, and categorical representation theory.

1. Perverse t-Structure: Definition and Key Properties

Let $X$ be a complex algebraic variety (or more generally, a suitably stratified topological space), and $D^b_c(X; k)$ the bounded derived category of constructible sheaves with coefficients in a field $k$. Fix a stratification $\{S_\alpha\}$ and a perversity function $p: \{\text{strata}\} \to \mathbb{Z}$, typically taking $p(S_\alpha) = -\operatorname{codim}(S_\alpha)$ for the middle perversity. The perverse t-structure is defined by:

$${}^pD^{\leq 0}(X) = \left\{ K \in D^b_c(X) \mid \forall \alpha, \forall k > p(S_\alpha): H^k(i_\alpha^* K) = 0 \right\}$$

$${}^pD^{\geq 0}(X) = \left\{ K \in D^b_c(X) \mid \forall \alpha, \forall k < p(S_\alpha): H^k(i_\alpha^! K) = 0 \right\}$$

The heart $\operatorname{Perv}(X)$ of the perverse t-structure is the abelian category ${}^pD^{\leq 0} \cap {}^pD^{\geq 0}$, whose objects are perverse sheaves. These categories are noetherian, artinian, and stable under Verdier duality. In the presence of odd-dimensional strata, the construction adapts using $p_-(S) = -\lfloor (\dim S + 1)/2 \rfloor$ and $p_+(S) = -\lfloor \dim S / 2 \rfloor$ (Saper, 2016).

Perverse sheaves satisfy vanishing conditions on cohomology for both stalks ($i_x^*$) and costalks ($i_x^!$), encoding intricate local information about the stratification and singularities of $X$ (Liu et al., 2019, Bhatt et al., 2023).

2. Algebraic, Analytic, and ℓ-adic Frameworks

The theory extends naturally between various coefficient and topological settings:

• Complex algebraic and analytic varieties: Perv(X) is defined in $D^b_c(X^{an}, \mathbb{C})$, with support/cosupport and codimension formulas (Bhatt et al., 2023).
• Étale and $\ell$-adic sheaves: Over excellent schemes, especially in dimension $\leq 2$, the perverse t-structure descends to the subcategory of Artin $\ell$-adic complexes $D^A(S, \mathbb{Z}_\ell)$, where objects are generated under extensions by pushforward from finite morphisms (2205.07796). The heart $\operatorname{Perv}^A(S, \mathbb{Z}_\ell)$ consists of complexes whose stalks are of Artin origin.
• Derived categories and D-modules: Under the Riemann–Hilbert correspondence, regular holonomic $\mathcal{D}$-modules correspond exactly to perverse sheaves (Bhatt et al., 2023, Kuwagaki, 2018).
• Infinite-dimensional categorical settings: Developments include perverse t-structures on stacks and ind-schemes, where gluing arguments and perverse stratifications extend via atlas procedures and stratified limit-gluing techniques (Bouthier et al., 2020).

3. Explicit Descriptions and Combinatorial Models

Perverse sheaves admit concrete combinatorial presentations in certain geometric contexts:

• Surfaces and Riemann surfaces: On a surface $S$ with singularities at $N \subset S$, perverse sheaves correspond exactly to double representations of the cell category $C_K$ of a spanning ribbon graph $K$ embedded in $S$. The combinatorial category $A_K$ encodes local systems on vertices and edges, together with gluing data satisfying cyclic relations reflecting the ribbon structure (Kapranov et al., 2016). A paracyclic category structure and Milnor disk formalism further enables an intrinsic abelian description of $\operatorname{Perv}(X, N)$ without reference to triangulated categories or t-structures (Dyckerhoff et al., 2020).
• Stratified spaces with finitely many strata: The category of perverse sheaves is equivalent to modules over a finite-dimensional algebra when each stratum’s category of local systems is finite and has enough projectives. Projective covers for simple perverse sheaves are constructed via recollement and explicit cone constructions, enabling algebraization (Cipriani et al., 2020).
• Symmetric products and Hilbert schemes: For the symmetric product $S^n(\mathbb{C}^2)$, perverse sheaves with respect to the collision stratification form an abelian category equivalent to modules over a Hilbert–Schur algebra $A_n$, generalizing Schur algebra structures and Springer theory (Braden et al., 2022).

4. Riemann–Hilbert Correspondence and D-module Equivalence

In both regular and irregular contexts, perverse sheaves are the topological counterparts of holonomic $\mathcal{D}$-modules:

• Regular Riemann–Hilbert: The de Rham functor $$dR : D^b_{rh}(\mathcal{D}_X) \to D^b_c(X^{an}, \mathbb{C})$$ is t-exact and identifies the perverse t-structure on sheaves with the standard t-structure on holonomic modules (Bhatt et al., 2023).
• Irregular Riemann–Hilbert: For $\mathbb{C}$-constructible sheaves with coefficients in a finite Novikov ring $\Lambda$, and with special $\mathbb{R}$-gradings, the bounded derived category $D^b_{ic}(\Lambda_X)$ of irregular constructible complexes is equivalent, via a modified D'Agnolo–Kashiwara correspondence, to the bounded derived category of holonomic (possibly irregular) $\mathcal{D}$-modules (Kuwagaki, 2018). The irregular perverse t-structure generalizes the usual one; its heart coincides with holonomic $\mathcal{D}$-modules.

Under these correspondences, simple objects in the abelian heart correspond to (possibly exponential) holonomic $\mathcal{D}$-modules, and the categorical equivalence commutes with all six Grothendieck operations.

5. Tannakian Formalism, Tensor Structure, and Applications

Perverse sheaves inherit tensor product structures and Tannakian formalism, essential for representation theory and moduli problems:

• Convolution product: Group actions, especially on semi-abelian varieties, induce convolution products that localize perverse sheaves to rigid symmetric monoidal abelian categories. Negligible objects (of Euler characteristic zero) are factored out, yielding pure tensor product structures (Krämer, 2013).
• Tannaka groups: For finitely tensor-generated subcategories, generic character twists yield fiber functors realizing the perverse sheaf category as representations of pro-algebraic groups. This connects perverse sheaf theory with moduli of local systems and geometric Langlands theory (Krämer, 2013).
• Generic vanishing and jump loci: On abelian and semi-abelian varieties, the cohomology jump loci $V^i(G,F)$ are finite unions of linear subvarieties, with propagation and codimension bounds characterizing perverse objects. Codimension conditions and generic vanishing provide explicit topological constraints and applications to duality spaces (Liu et al., 2019).
• Nearby cycles and degenerations: Tensoriality and degenerative behavior are controlled via nearby cycles, with Tannaka groups specializing under the nearby cycles functor, and weight filtration reflecting Hodge-theoretic and monodromic data (Krämer, 2013).

6. Algebraic Structures and Resurgence Theory

Perverse sheaves manifest rich algebraic and analytic structures in motivic, representation-theoretic, and physical contexts:

• Resurgence and wall-crossing: The category $\operatorname{Perv}(\mathbb{C}, A)$ of perverse sheaves with possibly infinite discrete sets of singularities admits a monoidal structure by additive convolution, relevant to resurgence theory and the calculus of resurgent functions (alien derivations, Stokes automorphisms) (Kapranov et al., 27 Dec 2025).
• Cohomological Hall Algebras and wall-crossing formulas: The assignment of Lefschetz perverse sheaves of vanishing cycles at critical values encodes algebraic structures for Cohomological Hall algebras and wall-crossing formula categorifications, whose convolution recovers the algebra product (Kapranov et al., 27 Dec 2025).
• Extension of classical invariants: Under Riemann–Hilbert and perverse theory, invariants such as Lyubeznik numbers, Bass numbers, and local cohomological dimension become embedding-independent and governed by the perverse cohomology at singular points. Strong Artin vanishing theorems and Cohen–Macaulay properties are proven for absolute integral closures and related singularities (Bhatt et al., 2023).

7. Extensions, Gluing, and Stack Formalism

Elementary gluing constructions and extensions of perverse sheaves facilitate localization and categorical equivalences:

• Quasi-inverse functor pairs: On Thom–Mather spaces with a closed stratum, the category of perverse sheaves admits an explicit description as data of a perverse sheaf on the open stratum, a local system on the stratum, together with gluing maps—making use of a "perverse closed set" [Editor's term]—and an explicit pair of quasi-inverse functors for the equivalence (Dupont, 2011).
• Combinatorial gluing and stack descent: The assignment of ribbon-graph–indexed categories constructs stacks of abelian categories with exact descent, allowing fully faithful realization of perverse sheaves by combinatorial objects locally and globally (Kapranov et al., 2016).
• Intrinsic abelian descriptions: Milnor–sheaf formalisms using paracyclic categories on surfaces recover perverse sheaves as abelian functors, bypassing derived categories and making duality operations explicit as categorical involutions (Dyckerhoff et al., 2020).

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Perverse sheaves constitute a unifying formalism that interrelates singularity theory, representation categories, D-module theory, and topological invariants. Their extensions to irregular, ℓ-adic, and categorical settings, algebraic models, tensor structures, and gluing mechanisms enable broad applications spanning arithmetic geometry, motivic theory, and higher categories. The recent fusion with resurgence theory, categorical wall-crossing, and advanced algebraic structures reflects ongoing research directions and deepening significance in pure mathematics and mathematical physics.