?W H A T I S . . .

a Perverse Sheaf?Mark Andrea de Cataldo and Luca Migliorini

Manifolds are obtained by gluing open subsets ofEuclidean space. Differential forms, vector fields,etc., are defined locally and then glued to yielda global object. The notion of sheaf embodiesthe idea of gluing. Sheaves come in many flavors:sheaves of differential forms, of vector fields,of differential operators, constant and locallyconstant sheaves, etc. A locally constant sheaf(local system) on a space X is determined byits monodromy, i.e., by a representation of thefundamental group π1(X, x) into the group ofautomorphisms of the fiber at x ∈ X: the sheafof orientations on the Möbius strip assigns −Idto the generators of the fundamental group Z. Asheaf, or even a map of sheaves, can be gluedback together from its local data: exterior deriva-tion can be viewed as a map between sheaves ofdifferential forms; the gluing is possible becauseexterior derivation is independent of the choice oflocal coordinates.

The theory of sheaves is made more completeby considering complexes of sheaves. A complexof sheaves K is a collection of sheaves {Ki}i∈Zand maps di : Ki → Ki+1 subject to d2 = 0. Thei-th cohomology sheafH i(K) is kerdi/imdi−1. The(sheafified) de Rham complex E is the complexwith entries the sheaves E i of differential i-formsand with differentials d : E i → E i+1 given by theexterior derivation of differential forms. By thePoincaré lemma, the cohomology sheaves are allzero, except for H 0 ≃ C, the constant sheaf.

The de Rham theorem, stating that the coho-mology of the constant sheaf equals closed formsmodulo exact ones, points to the fact that C and Eare cohomologically indistinguishable from eachother, even at the local level. The need to identifytwo complexes containing the same cohomologi-cal information via an isomorphism leads to the

Mark Andrea de Cataldo is professor of mathematics at

Stony Brook University. His email address is mde@math.

sunysb.edu.

Luca Migliorini is professor of geometry at the University

of Bologna. His email address is migliori@dm.unibo.it.

notion of derived category ([2]): the objects arecomplexes and the arrows are designed to achieve

the desired identifications. The inclusion of com-

plexes C ⊆ E is promoted by decree to the rank of

isomorphism in the derived category because it in-duces an isomorphism at the level of cohomology

sheaves.

While the derived category brings in a thicklayer of abstraction, it extends the reach and flex-

ibility of the theory. One defines the cohomology

groups of a complex and extends to complexes

of sheaves the ordinary operations of algebraictopology: pull-backs, push-forwards, cup and cap

products, etc. There is also a general form of

duality for complexes ([2]) generalizing classicalPoincaré duality.

Perverse sheaves live on spaces with singulari-

ties: analytic spaces, algebraic varieties, PL spaces,

pseudo-manifolds, etc. For ease of exposition, welimit ourselves to sheaves of vector spaces on com-

plex algebraic varieties and to perverse sheaves

with respect to what is called middle perversity.In order to avoid dealing with pathologies such

as sheaves supported on the Cantor set, one im-

poses a technical condition called constructibility.

Let us just say that the category DX of boundedconstructible complexes of sheaves on X sits in the

derived category and is stable under the various

topological operations mentioned above. If K is inDX , only finitely many of its cohomology sheaves

are nonzero and, for every i, the set suppH i(K),

the closure of the set of points at which the stalk

is nonzero, is an algebraic subvariety.A perverse sheafonX is a boundedconstructible

complex P ∈ DX such that the following holds for

K = P and for its dual P∨:

(1) dimC suppH −i(K) ≤ i, ∀i ∈ Z.

A map of perverse sheaves is an arrow in DX .

The term “sheaf” stems from the fact that,just as in the case of ordinary sheaves, (maps of)

perverse sheavescan be glued; as to “perverse”, see

below. The theory of perverse sheaves has its roots

632 Notices of the AMS Volume 57, Number 5

in the two notions of intersection cohomology andof D-module. As we see below, perverse sheavesandD-modules are related by the Riemann-Hilbertcorrespondence.

It is time for examples. If X is nonsingular,then CX[dimX], i.e., the constant sheaf in degree−dimCX, is self-dual and perverse. If Y ⊆ X isa nonsingular closed subvariety, then CY [dimY],viewed as a complex on X, is a perverse sheafon X. If X is singular, then CX[dimX] is usuallynot a perverse sheaf. On the other hand, theintersection cohomology complex (see below) isa perverse sheaf, regardless of the singularitiesof X. The extension of two perverse sheaves is aperverse sheaf. The following example can serveas a test case for the first definitions in the theoryof D-modules. Let X = C be the complex line withorigin o ∈ X, let z be the standard holomorphiccoordinate, let OX be the sheaf of holomorphicfunctions on X, let a be a complex number, andlet D be the differential operator D : f ֏ zf ′ − af .The complex Pa

(2) 0 -→ P−1a := OX

D-→ P 0

a := OX -→ 0

is perverse. If a ∈ Z≥0, then H −1(Pa) = CX andH 0(P0) = Co. If a ∈ Z<0, then H −1(Pa) is theextension by zero at o of the sheaf CX\o andH 0(Pa) = 0. If a ∉ Z, then H −1(Pa) is the exten-sion by zero at o of the local system on X \ oassociated with the branches of the multivaluedfunction za and H 0(Pa) = 0. In each case, the as-sociated monodromy sends the positive generatorof π1(X \ o,1) to e2πia. The dual of Pa is P−a (thisfits well with the notions of adjoint differentialequation and of duality for D-modules). Every Pais the extension of the perverse sheaf H 0(Pa)[0]by the perverse sheaf H −1(Pa)[1]. The extensionis trivial (direct sum) if and only if a ∉ Z>0.

A local system on a nonsingular variety canbe turned into a perverse sheaf by viewing it asa complex with a single entry in the appropri-ate degree. On the other hand, a perverse sheafrestricts to a local system on some dense opensubvariety. We want to make sense of the followingslogan: perverse sheaves are the singular versionof local systems. In order to do so, we discuss thetwo widely different ideas that led to the birthof perverse sheaves about thirty years ago: thegeneralized Riemann-Hilbert correspondence (RH)and intersection cohomology (IH) ([3]).

(RH) Hilbert’s 21st problem is concerned withFuchs-type differential equations on a puncturedRiemann surface Σ. As one circuits the punc-tures, the solutions are transformed: the sheaf ofsolutions is a local system on Σ (see (2)).

The 21st problem asked whether any local sys-tem arises in this way (it essentially does). Thesheafification of linear partial differential equa-tions on a manifold gives rise to the notion ofD-module. A regular holonomic D-module on a

complex manifold M is the generalization of theFuchs-type equations on Σ. The sheaf of solu-tions is now replaced by a complex of solutions,which, remarkably, belongs to DM . In (2), the com-plex of solutions is Pa, the sheaf of solutions toD(f ) = 0 is H −1(Pa), and H 0(Pa) is related to

the (non)solvability of D(f ) = g. Let Dbr,h(M) be

the bounded derived category of D-modules onM with regular holonomic cohomology. RH statesthat the assignment of the (dual to the) complexof solutions yields an equivalence of categories

Dbr,h(M) ≃ DM . Perverse sheaves enter the center

of the stage: they correspond via RH to regu-lar holonomic D-modules (viewed as complexesconcentrated in degree zero).

In agreement with the slogan mentioned above,the category of perverse sheaves shares the fol-lowing formal properties with the category of localsystems: it is Abelian (kernels, cokernels, images,and coimages exist, and the coimage is isomorphicto the image), stable under duality, Noetherian (theascending chain condition holds), and Artinian (thedescending chain condition holds), i.e., every per-verse sheaf is a finite iterated extension of simple(no subobjects) perverse sheaves. In our example,the perverse sheaves (2) are simple if and only ifa ∈ C \ Z.

What are the simple perverse sheaves? Inter-section cohomology provides the answer.

(IH) The intersection cohomology groups of asingular variety X with coefficients in a local sys-tem are a topological invariant of the variety. Theycoincide with ordinary cohomology whenX is non-singular and the coefficients are constant. Thesegroups were originally defined and studied usingthe theory of geometric chains in order to studythe failure, due to the presence of singularities, ofPoincaré duality for ordinary homology, and to puta remedy to it by considering the homology theoryarising by considering only chains that intersectthe singular set in a controlled way. In this context,certain sequences of integers, called perversities,were introduced to give a measure of how a chainintersects the singular set, whence the origin ofthe term “perverse”. The intersection cohomologygroups thus defined satisfy the conclusions ofPoincaré duality and of the Lefschetz hyperplanetheorem.

On the other hand, the intersection cohomologygroups can also be exhibited as the cohomologygroups of certain complexes in DX : the intersec-tion complexes of X with coefficients in the localsystem. It is a remarkable twist in the plot ofthis story that the simple perverse sheaves areprecisely the intersection complexes of the irre-ducible subvarieties of X with coefficients givenby simple local systems!

We are now in a position to clarify the ear-lier slogan. A local system L on a nonsingularsubvarietyZ ⊆M gives rise to a regular holonomic

May 2010 Notices of the AMS 633

D-module supported over the closure Z . The sameL gives rise to the intersection complex of Z withcoefficients in L. Both objects extend L from Zto Z across the singularities Z \ Z . By RH, theintersection complex is precisely the complex ofsolutions of the D-module.

A pivotal role in the applications of the theoryof perverse sheaves is played by the decompositiontheorem: let f : X → Y be a proper map of varieties;then the intersection cohomology groups of Xwith coefficients in a simple local system areisomorphic to the direct sum of a collection ofintersection cohomology groups of irreduciblesubvarieties of Y, with coefficients in simple localsystems. For example, if f : X → Y is a resolutionof the singularities of Y , then the intersectioncohomology groups of Y are a direct summandof the ordinary cohomology groups of X. This“as-simple-as-possible” splitting behavior is thedeepest known fact concerning the homology ofcomplex algebraic varieties and maps. It fails incomplex analytic and in real algebraic geometry.The decomposition of the intersection cohomologygroups ofX is a reflection in cohomology of a finerdecomposition of complexes in DY . The originalproof of the decomposition theorem usesalgebraicgeometry over finite fields (perverse sheaves makeperfect sense in this context). For a discussion ofsome of the proofs see [1].

One striking application of this circle of ideas isthe fact that the intersection cohomology groupsof projective varieties enjoy the same classicalproperties of the cohomology groups of projec-tive manifolds: the Hodge (p, q)-decompositiontheorem, the hard Lefschetz theorem, and theHodge-Riemann bilinear relations. This, of course,in addition to Poincaré duality and to the Lefschetzhyperplane theorem mentioned above.

The applications of the theory of perversesheaves range from geometry to combinatorics toalgebraic analysis. The most dramatic ones are inthe realm of representation theory, where theirintroduction has led to a truly spectacular revo-lution: proofs of the Kazhdan-Lusztig conjecture,of the geometrization of the Satake isomorphism,and, recently, of the fundamental lemma in theLanglands program (see the survey [1]).

References

[1] M. A. de Cataldo and L. Migliorini, The decompo-

sition theorem, perverse sheaves and the topology

of algebraic maps, Bulletin of the AMS 46 (2009),

535–633.

[2] L. Illusie, Catégories derivées et dualité, travaux de

J. L. Verdier, Enseign. Math. (2) 36 (1990), 369–391.

[3] S. Kleiman, The development of intersection ho-

mology theory, Pure and Applied Mathematics

Quarterly, 3 (2007), Special issue in honor of Robert

MacPherson, 225–282.

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