IV. Perverse Sheaves

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 22 October 2012

To any reader that has not met sheaves before: I suggest that you don't read this section, only refer to it a few times while you are reading Chapter VIII of these notes. The most important thing, from the point of view of these notes, is to understand the basic structures given in Chapter VIII; anyone who is going to study these topics in more depth can always come back and learn these definitions later.

A large part of the material in this section is basic material about derived categories. This material can usually be found in texts which treat homological algebra. Everything in this section, except the definition and properties of perverse sheaves given in §3 can be found in [KSc1980] Chapt. I-III. The definition of a perverse sheaf is in [BBD1992] 4.0 and the proof of Theorem (3.1) is in [BBD1992] Theorem 1.3.6. The Theorems in (3.2) are proved in [BBD1992] 2.1.9-2.1.11 and Theorem 4.3.1, respectively. We shall not review the definition of sheaves, it can be found in many textbooks, see [KSc1980] Chapt. II.

The category Dcb(X)

Complexes of sheaves

Let X be an algebraic variety. A complex of sheaves on X is a sequence of sheaves Ai on X and morphisms of sheaves di:AiAi+1,

A= ( d-2 A-1 d-1 A0 d0 A1 d1 ) such that di+1di=0.

The morphisms di:AiAi+1 are called the differentials of the complex A. Let A and B be complexes of sheaves. A morphism f:AB is a set of maps fn:AnBn such that the diagram

d-2 A-1 d-1 A0 d0 A1 d1 f-1 f0 f1 d-2 B-1 d-1 B0 d0 B1 d1


The ith cohomology sheaf of a complex A is the sheaf

i(A)= ker ( Ai Ai+1 ) im ( Ai-1 Ai )

We have a well defined complex of sheaves (A) given by

d-2 -1(A) d-1 0(A) d0 1(A) d1

A quasi-isomorphism f:AB is a morphism f:AB such that the induced morphism (f): (A) (B) is an isomorphism. Note that every isomorphism is a quasi-isomorphism but not the other way around (even though the notation may be confusing).

The category K(X) and derived functors

Let X be an algebraic variety. Let A and B be complexes of sheaves on X. Two morphisms f:AB and g:AB are homotopic is there is a collection of morphisms ki:AiBi-1 such that

fn-gn= kn+1dn+ dn-1kn.

The motivation for this definition is that if f and g are homotopic then (f)=(g).

Define K(X) to be the category given by

  1. Objects: Complexes of sheaves on X.
  2. Morphisms: A K(X)-morphism from a complex A to a complex B is an homotopy equivalence class of morphisms from A to B.

This just means that, in the category K(X), we identify homotopic morphisms.

Let A be a complex of sheaves on X. An injective resolution of A is a quasi-isomorphism AJ such that Ji is injective (an injective object in the category of sheaves on X) for all i. Let Sh(X) denote the category of sheaves on X and let F: Sh(X) Sh(X) be a functor. The right derived functor of G is the functor RF: K(X) K(X) given by

RF(A)=F(J)= ( F(d-2) F(J-1) F(d-1) F(J0) F(d0) F(J1) F(d1) )

where J is an injective resolution of A. The ith derived functor of F is the functor RiF: K(X) Sh(X) given by

RiF(A)= i(F(J)),

where J is an injective resolution of A. In other words RiF(A) is the ith cohomology sheaf of the complex RF(A).

Bounded complexes and constructible complexes

A complex of sheaves A is bounded if there exists a positive integer n such that Am=0 and A-m=0 for all m>n.

An algebraic stratification of an algebraic variety X is a finite partition X=αXα of X into strata such that

  1. For each α, the stratum Xα is a smooth locally closed algebraic subvariety in X,
  2. The closure of each stratum is a union of strata, and
  3. The whitney condition holds (see Verdier [Ver1976]).

Let l be a prime number and let l be the algebraic closure of the field l of l-adic numbers. A sheaf F on X is l-constructible if there is an algebraic stratification X=αXα such that, for each α, the restriction of F to Xα is a locally constant sheaf of finite dimensional vector spaces over l. A complex AK(X) is l-constructible if i(A) is l-constructible for all i.

Definition of the category Dcb(X)

Let X be a variety. Let A and B be complexes of sheaves on X. Define an equivalence relation on diagrams


in K(X) which have A and B as end points by saying that the diagram ACB, is equivalent to the diagram ACB, if there exists a commutative diagram


The notation CA denotes that the map is a quasi-isomorphism. The bounded derived category of l-constructible sheaves on X is the category Dcb(X) given by

  1. Objects: Bounded, l-constructible complexes of sheaves on X.
  2. Morphisms: A morphism from A to B is an equivalence class of diagrams ACB.

This definition of morphisms is a formal mechanism that inverts all quasi-isomorphisms. It ensures (in a coherent way) that "inverses" of quasi-isomorphisms are morphisms, i.e. that AB is a morphism from A to B.

Given two morphisms ADB and BEC in Dcb(X) one can show that there always exists a commutative diagram


and one defines the composition of the two morphisms ADB and BEC to be the morphism defined by the diagram AFC.


The direct image with compact support functor f!

A map g:XY between locally compact algebraic varieties is compact if the inverse image of every compact subset of Y is a compact subset of X.

Let f:XY be a morphism of locally compact algebraic varieties. Let F be a sheaf on X. The support, supps, of a section s of F on an open set V is the complement in V of the union of open sets UV such that sU=0.

The direct image with compact support sheaf f!F, is the sheaf on Y defined by setting

Γ(U;f!F)= { sΓ (f-1(U);F) f:suppsU is compact } ,

for every open set U in Y. (For a sheaf F on x and an open set U in X, Γ(U;F)= F(U).) This defines a functor f!: Sh(X) Sh(Y), where Sh(X) denotes the category of sheaves on X.

Let f:XY be a morphism of locally compact algebraic varieties. The direct image with compact support functor f!: Dcb(X) Dcb(Y) is given by


so that f! is the right derived functor of the functor f!: Sh(X) Sh(Y).

The inverse image functor f*

Let f:XY be a morphism of algebraic varieties. Let F be a sheaf on Y. The inverse image sheaf f*F is the sheaf on X associated to the presheaf

VlimUf(V) F(U), for allVopen inX,

where the limit is over all open sets U in Y which contain f(V). This defines a functor f*: Sh(Y) Sh(X).

The functor f

Let f:XY be a morphism of algebraic varieties. Let ADcb(X). Then fA is the unique (up to isomorphism) complex on Y such that


Actually, I have cheated here: We can only be sure that the complex fA is well defined if f is a locally trivial principal G-bundle, A is a semisimple G-equivariant complex on X and we require fA to be a semisimple complex on Y, see [Lus1993] 8.1.7 and 8.1.8 for definitions and details.

The shift functor [n]

Let A be a complex of sheaves on X. For each integer n define a new complex A[n], with differentials d[n]i, by

(A[n])i= An+i,and (d[n])i= (-1)ndn+i.

The shift functor is the functor

Dcb(X) [n] Dcb(X) A A[n].

The Verdier duality functor D

This definition is too involved for us to take the energy to repeat it here, we shall refer the reader to [KSc1980] §3.1. The main thing that we will need to know is that this functor exists.

Perverse sheaves

Definition of perverse sheaves

Let X be an algebraic variety. The support, suppF, of a sheaf F on X is the complement of the union of open sets UX such that FU=0.

A complex ADcb(X) is a perverse sheaf if

  1. dim suppi(A)=0 for i0 and dim suppi(A)-i for i<0, and
  2. dim suppi(D(A))=0 for i0 and dim suppi(D(A))-i for i<0,

where D(A) is the Verdier dual of A.

An abelian category is a category which has a direct sum operation and for which every morphism has a kernel and a cokernel. See [Ksc1980] I §1.2 for a precise definition.

The full subcategory of Dcb(X) whose objects are perverse sheaves on X is an abelian category.

Intersection cohomology complexes

Let YX be a smooth locally closed subvariety of complex dimensional d>0 and let be a locally constant sheaf on Y. There is a unique complex IC(Y,) in Dcb(X) such that

  1. i(IC(Y,)) =0, if i<-d,
  2. -d (IC(Y,)) Y=,
  3. dim suppi (IC(Y,)) -i, if i>-d,
  4. dim suppi (D(IC(Y,))) -i, if i>-d,

The complexes IC(Y,) are the intersection cohomology complexes and an explicit construction of these complexes is given in [BBD1992] Prop. 2.1.11.

The simple objects of the category of sheaves are the intersection complexes IC(Y,) as runs through the irreducible locally constant sheaves on various smooth locally closed subvarieties YX.

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

page history