A Perverse Sheaf? Mark Andrea De Cataldo and Luca Migliorini

WHAT IS... ? a Perverse Sheaf? Mark Andrea de Cataldo and Luca Migliorini

Manifolds are obtained by gluing open subsets of notion of derived category ([2]): the objects are Euclidean space. Differential forms, vector fields, complexes and the arrows are designed to achieve etc., are defined locally and then glued to yield the desired identifications. The inclusion of com- a global object. The notion of sheaf embodies plexes C ⊆ E is promoted by decree to the rank of the idea of gluing. Sheaves come in many flavors: isomorphism in the derived category because it in- sheaves of differential forms, of vector fields, duces an isomorphism at the level of cohomology of differential operators, constant and locally sheaves. constant sheaves, etc. A locally constant sheaf While the derived category brings in a thick (local system) on a space X is determined by layer of abstraction, it extends the reach and flex- its monodromy, i.e., by a representation of the ibility of the theory. One defines the cohomology fundamental group π1(X, x) into the group of groups of a complex and extends to complexes automorphisms of the fiber at x ∈ X: the sheaf of sheaves the ordinary operations of algebraic of orientations on the Möbius strip assigns −Id topology: pull-backs, push-forwards, cup and cap to the generators of the fundamental group Z.A products, etc. There is also a general form of sheaf, or even a map of sheaves, can be glued duality for complexes ([2]) generalizing classical back together from its local data: exterior deriva- Poincaré duality. tion can be viewed as a map between sheaves of Perverse sheaves live on spaces with singulari- differential forms; the gluing is possible because ties: analytic spaces, algebraic varieties, PL spaces, exterior derivation is independent of the choice of pseudo-manifolds, etc. For ease of exposition, we local coordinates. limit ourselves to sheaves of vector spaces on com- The theory of sheaves is made more complete plex algebraic varieties and to perverse sheaves by considering complexes of sheaves. A complex with respect to what is called middle perversity. i of sheaves K is a collection of sheaves {K }i∈Z In order to avoid dealing with pathologies such i i i+1 2 and maps d : K → K subject to d = 0. The as sheaves supported on the Cantor set, one im- H i i i−1 i-th cohomology sheaf (K) is ker d /im d . The poses a technical condition called constructibility. E (sheafified) de Rham complex is the complex Let us just say that the category D of bounded Ei X with entries the sheaves of differential i-forms constructible complexes of sheaves on X sits in the and with differentials d : Ei → Ei+1 given by the derived category and is stable under the various exterior derivation of differential forms. By the topological operations mentioned above. If K is in Poincaré lemma, the cohomology sheaves are all D , only finitely many of its cohomology sheaves zero, except for H 0 ≃ C, the constant sheaf. X are nonzero and, for every i, the set supp H i(K), The de Rham theorem, stating that the coho- the closure of the set of points at which the stalk mology of the constant sheaf equals closed forms is nonzero, is an algebraic subvariety. modulo exact ones, points to the fact that C and E A perverse sheaf on X is a bounded constructible are cohomologically indistinguishable from each complex P ∈ D such that the following holds for other, even at the local level. The need to identify X K = P and for its dual P ∨: two complexes containing the same cohomologi- −i cal information via an isomorphism leads to the (1) dimC supp H (K) ≤ i, ∀i ∈ Z.

Mark Andrea de Cataldo is professor of mathematics at A map of perverse sheaves is an arrow in DX . Stony Brook University. His email address is mde@math. The term “sheaf” stems from the fact that, sunysb.edu. just as in the case of ordinary sheaves, (maps of) Luca Migliorini is professor of geometry at the University perversesheavescan be glued; asto “perverse”,see of Bologna. His email address is [email protected]. 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 and complex manifold M is the generalization of the of D-module. As we see below, perverse sheaves Fuchs-type equations on Σ. The sheaf of solu- and D-modules are related by the Riemann-Hilbert tions is now replaced by a complex of solutions, correspondence. which, remarkably, belongs to DM . In (2), the com- It is time for examples. If X is nonsingular, plex of solutions is Pa, the sheaf of solutions to −1 0 then CX [dim X], i.e., the constant sheaf in degree D(f ) = 0 is H (Pa), and H (Pa) is related to b − dimC X, is self-dual and perverse. If Y ⊆ X is the (non)solvability of D(f ) = g. Let Dr,h(M) be a nonsingular closed subvariety, then CY [dim Y ], the bounded derived category of D-modules on viewed as a complex on X, is a perverse sheaf M with regular holonomic cohomology. RH states on X. If X is singular, then CX [dim X] is usually that the assignment of the (dual to the) complex not a perverse sheaf. On the other hand, the of solutions yields an equivalence of categories b intersection cohomology complex (see below) is Dr,h(M) ≃ DM . Perverse sheaves enter the center a perverse sheaf, regardless of the singularities of the stage: they correspond via RH to regu- of X. The extension of two perverse sheaves is a lar holonomic D-modules (viewed as complexes perverse sheaf. The following example can serve concentrated in degree zero). as a test case for the first definitions in the theory In agreement with the slogan mentioned above, of D-modules. Let X = C be the complex line with the category of perverse sheaves shares the fol- origin o ∈ X, let z be the standard holomorphic lowing formal properties with the category of local coordinate, let OX be the sheaf of holomorphic systems: it is Abelian (kernels, cokernels, images, functions on X, let a be a complex number, and and coimages exist, and the coimage is isomorphic let D be the differential operator D : f ֏ zf ′ − af . to the image), stable under duality, Noetherian (the The complex Pa ascending chain condition holds), and Artinian (the D descending chain condition holds), i.e., every per- (2) 0 -→ P −1 := O -→ P 0 := O -→ 0 a X a X verse sheaf is a finite iterated extension of simple ≥0 −1 is perverse. If a ∈ Z , then H (Pa) = CX and (no subobjects) perverse sheaves. In our example, 0 <0 −1 H (P0) = Co. If a ∈ Z , then H (Pa) is the the perverse sheaves (2) are simple if and only if extension by zero at o of the sheaf CX\o and a ∈ C \ Z. 0 −1 H (Pa) = 0. If a ∉ Z, then H (Pa) is the exten- What are the simple perverse sheaves? Inter- sion by zero at o of the local system on X \ o section cohomology provides the answer. associated with the branches of the multivalued (IH) The intersection cohomology groups of a a 0 function z and H (Pa) = 0. In each case, the as- singular variety X with coefficients in a local sys- sociated monodromy sends the positive generator tem are a topological invariant of the variety. They 2πia of π1(X \ o, 1) to e . The dual of Pa is P−a (this coincide with ordinary cohomology when X is non- fits well with the notions of adjoint differential singular and the coefficients are constant. These equation and of duality for D-modules). Every Pa groups were originally defined and studied using 0 is the extension of the perverse sheaf H (Pa)[0] the theory of geometric chains in order to study −1 by the perverse sheaf H (Pa)[1]. The extension the failure, due to the presence of singularities, of is trivial (direct sum) if and only if a ∉ Z>0. Poincaré duality for ordinary homology, and to put A local system on a nonsingular variety can a remedy to it by considering the homology theory be turned into a perverse sheaf by viewing it as arising by considering only chains that intersect a complex with a single entry in the appropri- the singular set in a controlled way. In this context, ate degree. On the other hand, a perverse sheaf certain sequences of integers, called perversities, restricts to a local system on some dense open were introduced to give a measure of how a chain subvariety. We want to make sense of the following intersects the singular set, whence the origin of slogan: perverse sheaves are the singular version the term “perverse”. The intersection cohomology of local systems. In order to do so, we discuss the groups thus defined satisfy the conclusions of two widely different ideas that led to the birth Poincaré duality and of the Lefschetz hyperplane of perverse sheaves about thirty years ago: the theorem. generalized Riemann-Hilbert correspondence (RH) On the other hand, the intersection cohomology and intersection cohomology (IH) ([3]). groups can also be exhibited as the cohomology st (RH) Hilbert’s 21 problem is concerned with groups of certain complexes in DX : the intersec- Fuchs-type differential equations on a punctured tion complexes of X with coefficients in the local Riemann surface Σ. As one circuits the punc- system. It is a remarkable twist in the plot of tures, the solutions are transformed: the sheaf of this story that the simple perverse sheaves are solutions is a local system on Σ (see (2)). precisely the intersection complexes of the irre- The 21st problem asked whether any local sys- ducible subvarieties of X with coefficients given tem arises in this way (it essentially does). The by simple local systems! sheafification of linear partial differential equa- We are now in a position to clarify the ear- tions on a manifold gives rise to the notion of lier slogan. A local system L on a nonsingular D-module.A regular holonomic D-module on a subvariety Z ⊆ M gives rise to a regular holonomic

May 2010 Notices of the AMS 633 D-module supported over the closure Z. The same L gives rise to the intersection complex of Z with coefficients in L. Both objects extend L from Z to Z across the singularities Z \ Z. By RH, the intersection complex is precisely the complex of solutions of the D-module. A pivotal role in the applications of the theory of perverse sheaves is played by the decomposition theorem: let f : X → Y be a proper map of varieties; then the intersection cohomology groups of X with coefficients in a simple local system are isomorphic to the direct sum of a collection of intersection cohomology groups of irreducible subvarieties of Y, with coefficients in simple local systems. For example, if f : X → Y is a resolution of the singularities of Y , then the intersection cohomology groups of Y are a direct summand of the ordinary cohomology groups of X. This “as-simple-as-possible” splitting behavior is the deepest known fact concerning the homology of complex algebraic varieties and maps. It fails in complex analytic and in real algebraic geometry. The decomposition of the intersection cohomology groups of X is a reflection in cohomology of a finer decomposition of complexes in DY . The original proof of the decompositiontheorem usesalgebraic geometry over finite fields (perverse sheaves make perfect sense in this context). For a discussion of some of the proofs see [1]. One striking application of this circle of ideas is the fact that the intersection cohomology groups of projective varieties enjoy the same classical properties of the cohomology groups of projective manifolds: the Hodge (p, q)-decomposition theorem, the hard Lefschetz theorem, and the Hodge-Riemann bilinear relations. This, of course, in addition to Poincaré duality and to the Lefschetz hyperplane theorem mentioned above. The applications of the theory of perverse sheaves range from geometry to combinatorics to algebraic analysis. The most dramatic ones are in the realm of representation theory, where their introduction 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 the Langlands program (see the survey [1]).

References [1] M. A. de Cataldo and L. Migliorini, The decomposition 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 homology theory, Pure and Applied Mathematics Quarterly, 3 (2007), Special issue in honor of Robert MacPherson, 225–282.

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