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Perverse sheaf

A perverse sheaf is a complex of sheaves on a topological space, typically an algebraic variety or stratified space, belonging to the heart of a specific t-structure on the bounded derived category of constructible sheaves; it satisfies support conditions on its cohomology sheaves, such as \dim \operatorname{supp} H^i(P) \leq -i and a dual condition on the Verdier dual \dim \operatorname{supp} H^i(P^\vee) \leq i for all i, ensuring stability under direct and inverse images as well as Verdier duality.^[1] These objects form an abelian subcategory, often denoted \operatorname{Perv}(X), which is Artinian and Noetherian, and they generalize local systems while providing a geometric realization for holonomic D-modules via the Riemann-Hilbert correspondence.^[2] Introduced by Alexander Beilinson, Joseph Bernstein, Pierre Deligne, and Ofer Gabber in their seminal 1982 work Faisceaux pervers, perverse sheaves were developed to resolve foundational issues in the study of singularities and stratifications in algebraic geometry.^[3] The motivation for perverse sheaves stems from the limitations of classical sheaf cohomology on singular spaces, where Poincaré duality fails; they provide a framework for intersection cohomology, originally defined topologically by Robert MacPherson and Mark Goresky in the late 1970s, by realizing it algebraically as the hypercohomology of the intersection complex \operatorname{IC}_X, a canonical perverse sheaf associated to a local system on the smooth part of X.^[4] This construction ensures that intersection cohomology satisfies key topological properties, such as homotopy invariance and the existence of a nondegenerate pairing, independent of the choice of Whitney stratification.^[1] Ofer Gabber provided crucial foundational contributions, including notes on t-structures that solidified the theory.^[5] Perverse sheaves exhibit remarkable functoriality: direct images under proper maps and inverse images under open immersions are t-exact, preserving the category, which enables powerful decomposition results.^[1] A cornerstone is the Beilinson-Bernstein-Deligne decomposition theorem, which states that for a proper map f: X \to Y between smooth varieties and a perverse sheaf P on X, the direct image Rf_* P semisimplely decomposes into a direct sum of shifted intersection complexes on the strata of Y, linking the topology of fibers to global invariants.^[4] This theorem has profound implications for understanding singularities and has been extended to mixed characteristic settings.^[1] In representation theory, perverse sheaves on flag varieties or the nilpotent cone categorify modules over Hecke algebras and Lie groups, yielding geometric interpretations of Kazhdan-Lusztig polynomials and standard modules via their Grothendieck groups.^[4] They also appear in the study of character sheaves and modular representations in positive characteristic.^[6] More broadly, perverse sheaves facilitate connections between geometry, topology, and analysis, such as in the study of Hodge modules and variations of Hodge structures on semi-abelian varieties.^[7] Their self-duality and vanishing theorems, like the Artin vanishing theorem (stating H^i(Y, P) = 0 for i > 0 and affine Y), underpin applications in enumerative geometry and mirror symmetry.^[1]

Foundations

Prerequisite Concepts

A sheaf of abelian groups on a topological space X begins with the notion of a presheaf \mathcal{F}, which assigns to each open set U \subseteq X an abelian group \mathcal{F}(U) of sections over U, together with restriction homomorphisms \rho_{U,V}: \mathcal{F}(U) \to \mathcal{F}(V) for V \subseteq U that satisfy compatibility conditions: \rho_{U,U} = \mathrm{id} and \rho_{V,W} \circ \rho_{U,V} = \rho_{U,W} for W \subseteq V \subseteq U.^[8] The sheaf condition requires that for any open cover \{U_i\}_{i \in I} of U, the following diagram is an equalizer: the map \mathcal{F}(U) \to \prod_i \mathcal{F}(U_i) equals the composition through pairwise intersections \prod_i \mathcal{F}(U_i) \rightrightarrows \prod_{i,j} \mathcal{F}(U_i \cap U_j).^[9] Stalks capture local behavior: for x \in X, the stalk \mathcal{F}_x is the direct limit \varinjlim_{U \ni x} \mathcal{F}(U) over neighborhoods U of x, consisting of germs of sections at x.^[8] Sections are elements of \mathcal{F}(U), with global sections \Gamma(X, \mathcal{F}) = \mathcal{F}(X). Morphisms of sheaves \phi: \mathcal{F} \to \mathcal{G} are collections of group homomorphisms \phi_U: \mathcal{F}(U) \to \mathcal{G}(U) commuting with restrictions, inducing stalk maps \phi_x: \mathcal{F}_x \to \mathcal{G}_x. The category of sheaves of abelian groups on X, denoted \mathrm{Sh}(X), is abelian, enabling the formation of complexes \cdots \to \mathcal{F}^i \to \mathcal{F}^{i+1} \to \cdots. The unbounded derived category D(X) is obtained by localizing the homotopy category of complexes at quasi-isomorphisms—chain maps inducing isomorphisms on cohomology H^i(\mathcal{F}^\bullet) = \ker(d^i)/\mathrm{im}(d^{i-1})—and is triangulated with shift functor

[\cdot]{{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}

and distinguished triangles. The bounded derived category D^b(X) \subset D(X) consists of complexes with bounded cohomology, i.e., H^i(\mathcal{F}^\bullet) = 0 for |i| \gg 0.^[10] For a continuous map f: X \to Y between topological spaces, the six-functor formalism provides derived functors on D(X) and D(Y): direct image f_* (right adjoint to inverse image f^*, both exact on sheaves), derived direct image Rf_* (right derived of f_*, computing higher direct images), extraordinary inverse image f^! (right adjoint to extraordinary direct image f_!, with f^! handling compact support via Verdier duality), and their derived versions. These satisfy base change, projection formulas, and purity for smooth maps.^[11] The bounded derived category D^b(X) plays a central role in cohomology computations: sheaf cohomology H^i(X, \mathcal{F}) is isomorphic to \mathrm{Hom}_{D(X)}(\mathbb{Z}_X, \mathcal{F}), where \mathbb{Z}_X is the constant sheaf, and hypercohomology of a complex \mathcal{K}^\bullet is H^i(X, \mathcal{K}^\bullet) = \mathrm{Hom}_{D(X)}(\mathbb{Z}_X, \mathcal{K}^\bullet), facilitated by resolutions and derived functors like R\Gamma(X, -).^[10] This framework unifies Čech, de Rham, and other cohomologies via triangulated structure.

Historical Context

Perverse sheaves were introduced in the early 1980s by Joseph Bernstein, Alexander Beilinson, and Pierre Deligne, in close collaboration with Ofer Gabber, as a central tool in the Riemann-Hilbert correspondence, which equates the category of regular holonomic D-modules on a complex manifold with the category of perverse sheaves of vector spaces solving the corresponding differential equations.^[12] This framework emerged from efforts to unify algebraic and analytic approaches to sheaf theory on singular spaces, providing a derived category structure that captures essential cohomological information. The primary motivation arose from intersection cohomology, pioneered by Mark Goresky and Robert MacPherson in the late 1970s, which sought a topological invariant for singular varieties that satisfies Poincaré duality without requiring resolution of singularities. Perverse sheaves formalized this through the concept of "middle perversity," offering a sheaf-theoretic extension functor that ensures well-behaved behavior in étale cohomology and resolves inconsistencies in direct image computations for singular morphisms.^[13] Significant influence came from microlocal sheaf theory, developed concurrently by Masaki Kashiwara and Pierre Schapira in the 1980s, which introduced microlocalization to study sheaves along singular supports and provided analytic tools for propagation of singularities that paralleled the algebraic perversity conditions. Key milestones include the 1982 IHÉS seminar notes "Faisceaux pervers" by Beilinson, Bernstein, and Deligne, which established the foundational t-structure and decomposition theorem. Later generalizations extended the theory to o-minimal structures, allowing perverse sheaves to apply in tame topological settings beyond classical algebraic geometry.^[14]

Definition

Core Definition

In the context of algebraic geometry, perverse sheaves are defined on a smooth stratified space X, such as a complex algebraic variety, equipped with a Whitney stratification. The underlying category is the bounded derived category of constructible sheaves D^b_c(X, \mathbb{Q}_\ell) (or more generally over a field of coefficients), where \mathbb{Q}_\ell denotes the \ell-adic rationals for \ell \neq \mathrm{char}(k) if X is defined over a field k.^[15] The perverse t-structure \mathbf{p} on D^b_c(X, \mathbb{Q}_\ell) is a specific t-structure defined relative to the stratification of X. For a stratum S \subset X with inclusion i: S \hookrightarrow X, an object K lies in \mathbf{p}D^{\leq 0} if, for every such i, the restriction i^* K has cohomology sheaves supported in degrees \leq -\dim S; dually, K lies in \mathbf{p}D^{\geq 0} if its Verdier dual D(K) lies in \mathbf{p}D^{\leq 0}, or equivalently, if i^! K has cohomology in degrees \geq -\dim S. This t-structure is generated by the truncation functors \tau^{\mathbf{p}}_{\leq m} and \tau^{\mathbf{p}}_{\geq m}, which are the right and left adjoints to the inclusions of \mathbf{p}D^{\leq m} and \mathbf{p}D^{\geq m}, respectively, and satisfy the standard axioms of a t-structure: the subcategories form a pair with no negative Ext groups between them, are stable under shift, and every object admits a distinguishing triangle.^[2] A perverse sheaf P is an object of D^b_c(X, \mathbb{Q}_\ell) belonging to the heart of the perverse t-structure, denoted \mathbf{p}\mathcal{H}^0 = {}^{\mathbf{p}}D^{\leq 0} \cap {}^{\mathbf{p}}D^{\geq 0}. The perverse cohomology functors are given by \mathbf{p}H^k = H^k \circ \tau^{\mathbf{p}}_{\geq k} \circ \tau^{\mathbf{p}}_{\leq k}. The heart {}^{\mathbf{p}}D^{\leq 0} \cap {}^{\mathbf{p}}D^{\geq 0} forms an abelian category, which is both Artinian and Noetherian, with short exact sequences corresponding to distinguished triangles in the t-structure.^[15]^[2]

Construction via Truncation

The construction of perverse sheaves often begins with standard complexes of sheaves in the bounded derived category D^b_c(X) of a variety X, using the truncation functors associated to the perverse t-structure. These functors, denoted ^p\tau_{\leq 0} and ^p\tau_{\geq 0}, project objects onto the subcategories ^pD^{\leq 0}_c(X) and ^pD^{\geq 0}_c(X), respectively, where the former consists of complexes F^\bullet satisfying \dim \operatorname{supp} H^j(F^\bullet) \leq -j for all j, and the latter is defined dually via the Verdier dual.^[16] Specifically, ^p\tau_{\leq 0} is the right adjoint to the inclusion ^pD^{\leq 0}_c(X) \hookrightarrow D^b_c(X), while ^p\tau_{\geq 0} is the left adjoint to the inclusion ^pD^{\geq 0}_c(X) \hookrightarrow D^b_c(X).^[17] For any complex F^\bullet \in D^b_c(X), the truncation functors fit into a canonical distinguished triangle

^p\tau_{\leq 0}(F^\bullet) \to F^\bullet \to ^p\tau_{\geq 1}(F^\bullet) \to (^p\tau_{\leq 0}(F^\bullet)){{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}},

which decomposes F^\bullet into components aligned with the perverse t-structure; this triangle, together with the dual triangle involving ^p\tau_{\geq 0} F^\bullet, allows the extraction of the zeroth perverse cohomology ^p\mathcal{H}^0(F^\bullet) = ^p\tau_{\leq 0} \circ ^p\tau_{\geq 0} (F^\bullet), which lies in \operatorname{Perv}(X).^[16]^[17] A key tool for constructing perverse sheaves on stratified spaces is the intermediate extension functor j_{! *}, which extends a perverse sheaf from an open dense stratum U \subset X (with inclusion j: U \hookrightarrow X and complement Z = X \setminus U) while preserving the perversity conditions. For Q^\bullet \in \operatorname{Perv}(U), j_{! *} Q^\bullet is defined as the image of the natural adjunction morphism ^p H^0(j_! Q^\bullet) \to ^p H^0(R j_* Q^\bullet) in the category of perverse sheaves, ensuring that j_{! *} Q^\bullet|_U \cong Q^\bullet, with no nonzero subobjects or quotients supported on Z.^[18] On a stratified variety, an explicit formula arises via successive applications of the classical truncation functors: j_{! *} Q^\bullet = \tau_{\leq -1} R j_{1 *} \cdots \tau_{\leq -m} R j_{m *} Q^\bullet, where m = \dim U and the j_k are inclusions of successive strata.^[17] To satisfy the support and cosupport conditions of the perverse t-structure, a standard adjustment involves shifting by the dimension of the ambient space. For a smooth variety X of dimension n and constant sheaf \underline{\mathbb{C}}_X, the shifted complex \underline{\mathbb{C}}_X is perverse because its cohomology is concentrated in degree -n with support of dimension n \leq n, and the dual condition holds via Verdier duality.^[16] More generally, for a local system L on a smooth open U \subset X, the intersection cohomology complex is given by \operatorname{IC}_X(L) = j_{! *}(L [\dim U]), which uses the shift to align with the perversity after extension.^[18] In the Riemann-Hilbert correspondence, which equates holonomic \mathcal{D}-modules with perverse sheaves of vanishing cycles, the perverse truncation of a complex F^\bullet can be expressed via the "real" and "imaginary" parts relative to the solution functor: the real part corresponds to the truncation ^p\tau_{\leq 0}(F^\bullet) capturing subanalytic supports, while the imaginary part arises in the distinguished triangle as the connecting term to

^p\tau_{\geq 1}(F^\bullet){{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}

, ensuring the middle cohomology sheaf is perverse.^[17]

Examples

Simple Geometric Examples

One of the simplest examples of a perverse sheaf arises on a smooth manifold X of dimension d. The constant sheaf \underline{k}_X shifted by the dimension, denoted \underline{k}_X, satisfies the conditions of the perverse t-structure, placing its cohomology sheaves in the appropriate degrees relative to supports.^[2] This shift ensures that the hypercohomology is concentrated such that the object lies in the heart of the perverse category.^[19] A classic mildly singular example is the intersection cohomology sheaf \mathrm{IC}_X on a cone X, such as the quadric cone C_n over a smooth projective variety of dimension n-1. Here, \mathrm{IC}_{C_n} = j_{!*} (\mathbb{Q}_U ), where U = C_n \setminus \{0\} is the open dense smooth stratum and j: U \hookrightarrow C_n is the inclusion.^[20] Away from the vertex, the stalks of \mathrm{IC}_{C_n} on U are those of the constant sheaf \mathbb{Q}_U, reflecting the smooth geometry there, while at the vertex, the stalk cohomology is concentrated in degree -n for the constant component, with additional vanishing or constant contributions depending on whether n is even or odd to satisfy perversity.^[20] Consider the inclusion i: \{p\} \hookrightarrow X of a point p into a space X. The pushforward i_* \underline{k}_{\{p\}} of the skyscraper sheaf at p is a perverse sheaf, as its support is the codimension-\dim X stratum \{p\}, and the stalk cohomology is \underline{k} in degree 0 at p with vanishing elsewhere.^[2] This verifies the perversity conditions, since the support dimensions align with the required inequalities for the t-structure: cohomology vanishes outside degrees compatible with the point's dimension 0.^[20] On the projective space \mathbb{P}^n, the constant sheaf \underline{k}_{\mathbb{P}^n} becomes a perverse sheaf after shifting by the dimension n, i.e., \underline{k}_{\mathbb{P}^n} .^[19] This follows from the smoothness of \mathbb{P}^n, where the coherent cohomology aligns with the perverse t-structure via truncation, placing the object in the heart.^[2]

Sheaves on Singular Varieties

Perverse sheaves are particularly well-suited for Whitney-stratified spaces, where a space X is decomposed into smooth strata S_\lambda satisfying Whitney's conditions (a) and (b), ensuring that the stratification is locally trivial and the tangent planes to higher strata limit properly to those of lower strata.^[20] On such spaces, a complex F^\bullet of constructible sheaves is perverse if, for each stratum S of dimension d, the cohomology sheaves H^i(F^\bullet) satisfy support and cosupport conditions relative to the stratum dimension: specifically, \dim \operatorname{supp} H^{-i}(F^\bullet) \leq i and \dim \operatorname{cosupp} H^i(F^\bullet) \leq -i, where the support is the locus where the sheaf is nonzero and the cosupport is the locus where the exceptional inverse image vanishes.^[21] These conditions ensure that the cohomology of F^\bullet is concentrated in degrees compatible with the stratification, allowing perverse sheaves to capture topological features across singular loci without excessive growth in cohomology dimensions.^[20] Deligne's construction of the intersection cohomology complex \mathrm{IC}^\bullet_X proceeds iteratively along the strata of a Whitney stratification of X. Starting with a local system L on the top-dimensional open stratum U, one pushes forward L[\dim U] to X and applies the perverse truncation functor \tau_{\leq 0} to obtain a complex supported on the closure of U; this process is repeated for lower strata, taking intermediate extensions (the image of the adjunction map from the truncation to the pushforward) to ensure the resulting complex is perverse and extends uniquely.^[21] The intermediate extension functor \mathrm{IC} preserves the local system on smooth parts while controlling behavior near singularities, yielding \mathrm{IC}^\bullet_X(L) as the unique perverse sheaf satisfying these gluing conditions across the stratification.^[20] A concrete example arises on a nodal curve, such as the singular fiber in the Weierstrass family y^2 = x(x-a)(x-b) over \mathbb{C}, where the perverse sheaf associated to the pushforward

f_* \mathbb{Q}_E{{grok:render&&&type=render_inline_citation&&&citation_id=2&&&citation_type=wikipedia}}

(with E the smooth total space) decomposes into semisimple summands, including the intersection complex on the nodal stratum and shifts of simple perverse sheaves supported at the node.^[20] The hypercohomology \mathbb{H}^*(X, \mathrm{IC}^\bullet_X) then computes the intersection cohomology groups, which for this nodal elliptic curve yield Betti numbers reflecting the genus of the smooth model (e.g., b_1 = 1) while the vanishing cycles at the node capture the monodromy action, distinguishing the singular topology from the smooth case.^[20] In the context of resolution of singularities, perverse sheaves detect vanishing cycles by relating the pushforward from a resolution \pi: \tilde{X} \to X to the base space. Specifically, \pi_* \mathbb{Q}_{\tilde{X}}[\dim X] is perverse on X, and the vanishing cycle functor \psi_\pi extracts the kernel of the map from the nearby cycles to the stalk on the base, measuring the cohomological drop across singular fibers; for instance, on a resolution of a nodal curve, these cycles are supported at the exceptional divisor and isomorphic to skyscraper sheaves encoding the singularity type.^[21] This allows perverse sheaves to encode the failure of Poincaré duality on singular varieties, with the decomposition theorem splitting the pushforward into intersection complexes plus corrections from vanishing cycles.^[20]

Properties

t-Structure and Heart

The perverse t-structure on the bounded derived category of constructible sheaves D^b_c(X) on a variety X is defined by a pair of full subcategories ^pD^{\leq 0}(X) and ^pD^{\geq 0}(X) satisfying specific axioms that ensure compatibility with the triangulated structure. Specifically, the axioms include orthogonality: \Hom(A, B) = 0 for all i > 0 when A \in ^pD^{\leq 0}(X) and B \in ^pD^{\geq 0}(X); inclusion properties: ^pD^{\leq 0}(X) \subseteq ^pD^{\leq 1}(X) and ^pD^{\geq 0}(X) \supseteq ^pD^{\geq -1}(X); and existence of truncation functors, where for any complex K \in D^b_c(X), there is a distinguished triangle ^p\tau^{\leq 0} K \to K \to ^p\tau^{\geq 1} K \to with ^p\tau^{\leq 0} K \in ^pD^{\leq 0}(X) and ^p\tau^{\geq 1} K \in ^pD^{\geq 1}(X).^[2] These truncation functors are the right and left adjoints to the inclusion functors, respectively, and the perverse cohomology functors are given by

^p\mathcal{H}^i(K) = ^p\tau^{\leq i} ^p\tau^{\geq i} K

.^[2] For the middle perversity, the subcategories are defined relative to a Whitney stratification of X, with K \in ^pD^{\leq 0}(X) if for every stratum S, the restriction to S has cohomology vanishing above degree -\dim S, and K \in ^pD^{\geq 0}(X) if for every stratum S, the exceptional inverse image to S has cohomology vanishing below degree -\dim S; this definition is independent of the choice of stratification.^[15] The heart of the perverse t-structure, denoted \mathrm{Perv}(X), is the full subcategory ^pD^{\leq 0}(X) \cap ^pD^{\geq 0}(X) consisting of perverse sheaves, which are complexes K satisfying the support and cosupport conditions for all strata. This heart forms an abelian category, as it is closed under extensions, kernels, and cokernels, with the short exact sequences arising from distinguished triangles in the derived category. Moreover, \mathrm{Perv}(X) is both Artinian and Noetherian, ensuring that every object has finite length.^[2] The six functor formalism is compatible with the perverse t-structure, meaning that the functors f_!, f_*, f^*, f^!, \otimes^L, and \mathbb{R}\mathrm{Hom} are t-exact or have controlled amplitude relative to it. For a morphism f: X \to Y of varieties, if f is affine, then f_* is right t-exact and f^! is left t-exact; if the fibers of f have dimension at most d, then f_! and f^* have amplitude [0, d], while f_* and f^! have amplitude [-d, 0]. For smooth morphisms of relative dimension d, f^* and f^! are t-exact isomorphisms; Verdier duality D is t-exact on the derived category.^[2] The external tensor product \boxtimes is t-exact, and the left-derived tensor \otimes^L is left t-exact.^[2] On quasi-projective varieties over a field, the heart \mathrm{Perv}(X) forms a coherent abelian category, meaning it has enough projectives and injectives with coherent Hom-spaces, which follows from the Noetherian and Artinian properties combined with the existence of a dualizing complex.^[22] This coherence ensures that perverse sheaves behave well under gluing and form a stack over the étale or analytic site.^[2]

Duality and Self-Duality

In the derived category of constructible sheaves D^b_c(X) on a variety X, Verdier duality provides an equivalence D: D^b_c(X) \to D^b_c(X)^{op} given by D(K) = \RHom(K, \omega_X), where \omega_X is the dualizing complex, and it satisfies D^2 \cong \id. This duality preserves the subcategory of bounded perverse sheaves, mapping the heart of the middle perversity t-structure to itself, thus endowing the category of perverse sheaves with a natural contravariant equivalence.^[20] For a morphism f: X \to Y, Verdier duality interchanges direct and inverse images via the relation f^! \dashv Rf_*, where f^! is the right adjoint to the derived direct image Rf_*, ensuring that duality respects the geometry of maps between spaces.^[18] For a perverse sheaf P on a smooth variety X of dimension d, the explicit formula for its Verdier dual is D(P) \cong \RHom(P, \omega_X), which lies in the heart of the perverse t-structure and preserves perversity conditions.^[18] In particular, if P = L where L is a local system on an irreducible smooth component of dimension d, then D(P) \cong L^\vee [d](d), with L^\vee the dual local system and (d) the Tate twist.^[18] The intersection cohomology complex \IC_X on a variety X of dimension n exhibits self-duality under Verdier duality, satisfying \IC_X \cong D(\IC_X), which preserves the middle perversity condition and reflects the Poincaré duality inherent in intersection cohomology. This isomorphism ensures that the simple perverse sheaves arising from intersection cohomology are rigid under duality.^[17] In the perverse setting, Artin vanishing asserts that for an affine variety A of dimension m and a perverse sheaf P \in {}^p D^{\leq 0}(A), the cohomology groups H^i(A, P) = 0 for i > 0, providing a cohomological bound analogous to the classical Artin vanishing for coherent sheaves.^[20] Similarly, the hard Lefschetz theorem extends to perverse sheaves on Kähler varieties, where for a class \eta \in H^2(X, \mathbb{Q}) of Hodge type (1,1), the operator of cupping with \eta^k induces isomorphisms H^{n-k}(X, \IC_X) \to H^{n+k}(X, \IC_X) for $0 \leq k \leq n, realized in the heart of the perverse t-structure via the decomposition theorem.^[20] Duality is compatible with direct images for perverse sheaves: if f: X \to Y is proper and P is a perverse sheaf on X, then D(Rf_*(P)) \cong f_*(D(P)), preserving the perverse category and enabling computations of global sections through dual local data.^[18] This relation follows from the adjunction f^! \dashv Rf_* and the fact that for proper maps, Rf_* = f_!.

Applications

In Algebraic Geometry

Perverse sheaves provide a powerful framework for computing intersection cohomology groups of singular algebraic varieties, enabling the study of their topological and geometric invariants. For a stratified variety X, the intersection cohomology complex \mathrm{IC}_X, known as the intermediate extension sheaf, is a simple perverse sheaf that captures the intersection homology in a way that satisfies Poincaré duality even for singular spaces. The intersection cohomology groups \mathrm{IH}^i(X) are then computed as the hypercohomology groups \mathbb{H}^i(X, \mathrm{IC}_X), which benefit from the t-structure on the derived category of perverse sheaves to control supports and cohomology degrees. This approach, introduced in the foundational work on perverse sheaves, allows for explicit calculations in cases like projective varieties with isolated singularities, where the stalks of \mathrm{IC}_X restrict to constant sheaves on smooth strata and vanish appropriately on lower-dimensional ones.^[3]^[23] The Riemann-Hilbert correspondence further bridges algebraic geometry with differential equations by establishing an equivalence between the category of regular holonomic \mathcal{D}_X-modules on a complex manifold X and the category of perverse sheaves on X with coefficients in the constant sheaf \mathbb{C}. This duality, proven independently by Kashiwara and Mebkhout, implies that solutions to holonomic differential systems correspond to geometric data encoded in perverse sheaves, facilitating the study of singularities in families of varieties through microlocal analysis. In particular, the solution complex of a regular holonomic \mathcal{D}-module is a perverse sheaf, and the correspondence preserves exactness and supports, allowing transfers of properties like purity between the two categories.^[24]^[25] More recently, perverse sheaves have been applied in commutative algebra through the Riemann-Hilbert correspondence to analyze properties like support conditions translating to Lyubeznik numbers and local cohomology of singular rings.^[26] Perverse sheaves are integral to the theory of mixed Hodge modules, developed by Saito, which extends Hodge theory to singular varieties by endowing perverse sheaves with compatible filtrations and weight structures. In this framework, mixed Hodge modules on a complex variety X are pairs consisting of a coherent \mathcal{D}_X-module and a perverse sheaf, linked via the Riemann-Hilbert correspondence, with nearby and vanishing cycle functors defined to handle degenerations along hypersurfaces. These functors, such as \psi_f for the vanishing cycles of a morphism f: X \to \mathbb{C}, produce perverse sheaves that encode the monodromy and limiting mixed Hodge structures on the cohomology of fibers, enabling computations of Hodge numbers for singular families like those arising in mirror symmetry or Calabi-Yau degenerations. The strictness of direct images under these functors ensures that the associated graded pieces remain perverse sheaves, preserving the abelian category structure.^[27]^[28] In applications to Hodge theory, perverse sheaves underpin the purity and decomposition theorems for semisimple complexes, providing a geometric realization of mixed Hodge structures on intersection cohomology. The purity theorem asserts that for a pure perverse sheaf of weight w, its cohomology sheaves are pure of weight w + i in degree i, ensuring that the hypercohomology carries a pure Hodge structure. The decomposition theorem, a cornerstone result, states that for a proper morphism f: X \to Y between smooth projective varieties and the constant perverse sheaf on X, the direct image R f_* \mathbb{Q}_X decomposes in the perverse t-structure as a direct sum of shifts of intersection cohomology sheaves on irreducible components of Y, each pure in the Hodge sense. This semisimplicity, combined with Verdier duality, implies that the perverse filtration on the cohomology of Y arises from a canonical splitting, with applications to computing Hodge-Deligne numbers for projective maps.^[3]^[23]

In Representation Theory and Physics

Perverse sheaves play a central role in representation theory through the geometric Satake equivalence, which establishes a canonical tensor equivalence between the category of perverse sheaves on the affine Grassmannian of a reductive algebraic group G and the category of representations of its Langlands dual group \hat{G}. Specifically, for a reductive group G over an algebraically closed field k of characteristic zero and the affine Grassmannian \mathrm{Gr}_G = G(k((t)))/G(k[]), the category P_{G(k[])}(\mathrm{Gr}_G, k) of G(k[])-equivariant perverse sheaves is equivalent to \mathrm{Rep}(\hat{G}_k), the category of finite-dimensional representations of the dual group scheme \hat{G}_k. This equivalence, proved using global sections as a fiber functor and Tannakian reconstruction, translates geometric constructions on the Grassmannian—such as convolution products—into tensor products of representations, providing a geometric realization of the Satake isomorphism for representations of complex groups on flag varieties.^[29] In microlocal sheaf theory, perverse sheaves extend to microlocal perverse sheaves on the cotangent bundle, capturing the propagation of singularities along bicharacteristic leaves. Microlocal perverse sheaves are defined as complexes of microlocal sheaves (ind-sheaves on the projective cotangent bundle P^*X of a manifold X) that are perverse with respect to the microlocal t-structure, with micro-supports that are C^\times-conic Lagrangian subvarieties satisfying non-characteristic conditions. The characteristic variety of a microlocal perverse sheaf, given by its micro-support \mathrm{SS}(F), determines the directions of singularity propagation: singularities propagate along the bicharacteristic relation unless obstructed by the perverse condition, which ensures holonomic behavior akin to that of perverse sheaves in the classical setting. This framework, developed by Kashiwara and Schapira, allows analysis of singularity propagation for solutions to PDEs via the Riemann-Hilbert correspondence, linking microlocal perverse sheaves to holonomic D-modules.^[30]^[31] Applications in physics, particularly string theory, arise through perverse sheaves in mirror symmetry and topological string models. In type IIB string theory, perverse sheaves realize intersection space cohomology for singular Calabi-Yau varieties, correctly counting massless 3-branes during conifold transitions, unlike intersection cohomology which counts 2-branes in type IIA. For projective hypersurfaces with isolated singularities, the intersection space complex is a self-dual perverse sheaf whose hypercohomology computes this modified cohomology, carrying a mixed Hodge structure and enabling Poincaré duality via Verdier self-duality. In mirror symmetry, perverse sheaves (or their microlocal variants) relate the B-model on the complex side to the A-model Fukaya category: for singular hypersurfaces, localization of the wrapped Fukaya category at the invariant cycles endofunctor yields a category equivalent to that of perverse sheaves on the mirror Landau-Ginzburg model, facilitating counting of invariants like Gromov-Witten numbers through homological mirror symmetry. This connection extends to the topological B-model, where perverse sheaves encode deformation-stable invariants for singular mirrors.^[32] Recent developments since 2000 have linked perverse sheaves to categorical actions and quantum groups via affine braid group actions on derived categories. In the context of modular representations, the affine braid group acts on the derived category of coherent sheaves on the Springer resolution, inducing categorical actions that mirror representations of affine Hecke algebras and quantum groups at roots of unity. Bezrukavnikov and Riche established such actions on categories involving perverse coherent sheaves on affine flag varieties, connecting to Lusztig's conjectures on canonical bases and providing t-structures compatible with quantum group symmetries. These constructions, building on geometric Satake, enable Koszul duality between categories of perverse sheaves and quantum group modules, with applications to positive characteristic representations.^[33]

References

[1]
https://arxiv.org/pdf/1506.03642.pdf
[2]
[PDF] Quick and dirty introduction to perverse sheaves - Math (Princeton)
This is a very quick introduction to the definition of perverse sheaves, that also tries to give some motivation (coming from the theory of D-modules).
[3]
[PDF] Faisceaux pervers
Les faisceaux pervers n'étant ni des faisceaux, ni pervers, la ter- minologie requiert une explication. Le mot "pervers". n'enchante pas certains de nous ...
[4]
https://arxiv.org/pdf/math/0307349.pdf
[5]
https://www.ihes.fr/~gabber/t-str.pdf
[6]
[0901.3322] Perverse sheaves and modular representation theory
Jan 21, 2009 · This paper is an introduction to the use of perverse sheaves with positive characteristic coefficients in modular representation theory.
[7]
[1902.05430] Perverse sheaves on semi-abelian varieties - arXiv
Feb 13, 2019 · We survey recent developments in the study of perverse sheaves on semi-abelian varieties. As concrete applications, we discuss various obstructions.
[8]
[PDF] A Very Elementary Introduction to Sheaves - arXiv
Mar 1, 2022 · For a topological space X, a sheaf of abelian groups on X assigns each open set of X to an abelian group. Which abelian group you ask? Here ...
[9]
Sheaf Cohomology - an overview | ScienceDirect Topics
Godement's 1958 book [79] summarized and refined all these developments, becoming the standard reference for sheaves, sheaf cohomology and spectral sequences ...
[10]
https://therisingsea.org/notes/DerivedCategoriesOfSheaves.pdf
[11]
[PDF] MASAKI KASHIWARA PIERRE SCHAPIRA Ind-sheaves - Numdam
May 3, 2023 · We construct "Grothendieck's six operations" in the derived categories of ind- sheaves, as well as new functors which naturally arise. There is ...
[12]
Preface
Perverse sheaves were discovered in the fall of 1980 by Beilinson–Bernstein–. Deligne–Gabber [24], sitting at the confluence of two major developments of ...
[13]
[PDF] Perverse sheaves and the topology of algebraic varieties
It was first proved by Beilinson-Bernstein-Deligne-. Gabber in their ... A brief history of perverse sheaves. Intersection complexes were invented by ...
[14]
[PDF] lecture notes on sheaves and perverse sheaves
Aug 9, 2021 · ... sheaves of individual D-modules. Thus was born the category of perverse sheaves, with middle perversity. On the other hand, intersection ...
[15]
[PDF] PERVERSE SHEAVES AND t-STRUCTURES - Purdue Math
Feb 27, 2013 · Thus we can define perverse sheaves of Q` sheaves by imitating ... Theorem 4.3 (Beilinson, Bernstein, Deligne). The heart is abelian ...
[16]
[PDF] 8 Perverse Sheaves - Purdue Math
Jan 5, 2020 · An obvious origin of the theory of perverse sheaves is the Riemann–Hilbert correspon- dence. Indeed, as we have seen in Section 7.2 one ...
[17]
[PDF] perverse sheaves - Cornell Mathematics
The goal of this course is to give an introduction to perverse sheaves, to prove the decomposition theorem and then to highlight several applications of ...
[18]
[PDF] perverse sheaves - Yale Math
Let f : X → Y be an affine morphism. Then f∗ is right t-exact (and f! is left t-exact) for the perverse t-structures. Combining ...
[19]
[PDF] Perverse Sheaves - UMD MATH
Jul 27, 2020 · Similarly, D(F[n]) = F∨[n] is a perverse sheaf. This includes constant sheaves and, over a nite eld, the Tate twisting sheaf Ql(1). Example 2.6.<|control11|><|separator|>
[20]
[PDF] An illustrated guide to perverse sheaves
Perverse sheaves provide a powerful tool for understanding the topol- ogy of algebraic varieties. They really come into their own when trying to understand the ...
[21]
[PDF] Lecture Notes on Sheaves and Perverse Sheaves
Aug 3, 2024 · Perverse sheaves. 66. 19. Perverse sheaves. 66. 20. Examples of perverse sheaves. 68. 21. t-structures and Perverse cohomology. 76. 22.
[22]
[PDF] NOTES ON SOME t-STRUCTURES - IHES
Feb 3, 2020 · Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, analyse et topologie sur les espaces singuliers, Astérisque 100, Soc. Math. France ...
[23]
[PDF] The decomposition theorem, perverse sheaves and the topology of ...
Apr 16, 2009 · Abstract. We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, ...
[24]
D-Modules, Perverse Sheaves, and Representation Theory
In stock Free deliveryKey to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory.
[25]
Riemann-Hilbert correspondence for holonomic D-modules - Numdam
In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular.
[26]
[PDF] Mixed Hodge Modules - Purdue Math
underlying perverse sheaf of its object is a local system shifted by dim U ... for ^ = ICX-QH by (4.4.1), because Hijlj~1J( = Q(i>Q) (resp. WjJ'1^. = 0 ...
[27]
Nearby and vanishing cycles for perverse sheaves and D-modules
Sep 28, 2017 · Abstract:We survey nearby and vanishing cycles for both perverse sheaves and D-modules under analytic setting.
[28]
[PDF] Microlocal study of sheaves - Numdam
rize perverse sheaves as those sheaves whose shift is zero (microlo- cally ... Kashiwara-Schapira []3 ) . In a similar way we give a bound to the.
[29]
[math/0209341] Microlocal perverse sheaves - arXiv
Sep 25, 2002 · Microlocal perverse sheaves will be represented as complexes of analytic ind-sheaves which have recently been studied by Kashiwara-Schapira.Missing: sheaf theory
[30]
Intersection spaces, perverse sheaves and type IIB string theory - arXiv
Dec 10, 2012 · ... duality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore ...
[31]
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