IV. Perverse Sheaves

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $D_c^b\left(X\right)$

Complexes of sheaves

Let $X$ be an algebraic variety. A complex of sheaves on $X$ is a sequence of sheaves $A^i$ on $X$ and morphisms of sheaves $d_i:A^i\to A^{i+1},$

$$A=\left(\dots \stackrel{d_{-2}}{⟶}{A}^{-1}\stackrel{d_{-1}}{⟶}{A}^0\stackrel{d_0}{⟶}{A}^1\stackrel{d_1}{⟶}\dots \right)\text{such that}{d}_{i+1}{d}_i=0\text{.}$$

The morphisms $d_i:A^i\to A^{i+1}$ are called the differentials of the complex $A\text{.}$ Let $A$ and $B$ be complexes of sheaves. A morphism $f:A\to B$ is a set of maps $f_n:A^n\to B^n$ such that the diagram

$$\begin{array}{ccccccccc}\dots & \stackrel{d_{-2}}{⟶}& A^{-1}& \stackrel{d_{-1}}{⟶}& A^0& \stackrel{d_0}{⟶}& A^1& \stackrel{d_1}{⟶}& \dots \\ & & \downarrow f_{-1}& & \downarrow f_0& & \downarrow f_1\\ \dots & \stackrel{d_{-2}}{⟶}& B^{-1}& \stackrel{d_{-1}}{⟶}& B^0& \stackrel{d_0}{⟶}& B^1& \stackrel{d_1}{⟶}& \dots \end{array}$$

commutes.

The $i\text{th}$ cohomology sheaf of a complex $A$ is the sheaf

$${\mathscr{H}}^i\left(A\right)=\frac{\text{ker}(A^i\to A^{i+1})}{\text{im}(A^{i-1}\to A^i)}$$

We have a well defined complex of sheaves $\mathscr{H}\left(A\right)$ given by

$$\dots \stackrel{d_{-2}}{⟶}{\mathscr{H}}^{-1}\left(A\right)\stackrel{d_{-1}}{⟶}{\mathscr{H}}^0\left(A\right)\stackrel{d_0}{⟶}{\mathscr{H}}^1\left(A\right)\stackrel{d_1}{⟶}\dots$$

A quasi-isomorphism $f:A\stackrel{\sim }{\to }B$ is a morphism $f:A\to B$ such that the induced morphism $\mathscr{H}\left(f\right):\mathscr{H}\left(A\right)\to \mathscr{H}\left(B\right)$ 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 $K\left(X\right)$ and derived functors

Let $X$ be an algebraic variety. Let $A$ and $B$ be complexes of sheaves on $X\text{.}$ Two morphisms $f:A\to B$ and $g:A\to B$ are homotopic is there is a collection of morphisms $k_i:A^i\to B^{i-1}$ such that

$$f_n-g_n=k_{n+1}{d}_n+d_{n-1}{k}_n\text{.}$$

The motivation for this definition is that if $f$ and $g$ are homotopic then $\mathscr{H}\left(f\right)=\mathscr{H}\left(g\right)\text{.}$

Define $K\left(X\right)$ to be the category given by

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

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

Let $A$ be a complex of sheaves on $X\text{.}$ An injective resolution of $A$ is a quasi-isomorphism $A\stackrel{\sim }{\to }J$ such that $J^i$ is injective (an injective object in the category of sheaves on $X\text{)}$ for all $i\text{.}$ Let $\text{Sh}\left(X\right)$ denote the category of sheaves on $X$ and let $F:\text{Sh}\left(X\right)\to \text{Sh}\left(X\right)$ be a functor. The right derived functor of $G$ is the functor $RF:K\left(X\right)\to K\left(X\right)$ given by

$$RF\left(A\right)=F\left(J\right)=(\dots \stackrel{F\left(d_{-2}\right)}{⟶}F\left(J^{-1}\right)\stackrel{F\left(d_{-1}\right)}{⟶}F\left(J^0\right)\stackrel{F(d_0)}{⟶}F\left(J^1\right)\stackrel{F(d_1)}{⟶}\dots )$$

where $J$ is an injective resolution of $A\text{.}$ The ith derived functor of $F$ is the functor $R^{i}F:K\left(X\right)\to \text{Sh}\left(X\right)$ given by

$$R^{i}F\left(A\right)={\mathscr{H}}^i\left(F\left(J\right)\right),$$

where $J$ is an injective resolution of $A\text{.}$ In other words $R^{i}F\left(A\right)$ is the $i\text{th}$ cohomology sheaf of the complex $RF\left(A\right)\text{.}$

Bounded complexes and constructible complexes

A complex of sheaves $A$ is bounded if there exists a positive integer $n$ such that $A^m=0$ and $A^{-m}=0$ for all $m>n\text{.}$

An algebraic stratification of an algebraic variety $X$ is a finite partition $X=\underset{\alpha }{⨆}{X}_{\alpha }$ of $X$ into strata such that

1. For each $\alpha ,$ the stratum $X_{\alpha }$ 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 $\overline{{\mathbb{Q}}_l}$ be the algebraic closure of the field ${\mathbb{Q}}_l$ of $l\text{-adic}$ numbers. A sheaf $F$ on $X$ is $\overline{{\mathbb{Q}}_l}\text{-constructible}$ if there is an algebraic stratification $X=\underset{\alpha }{⨆}{X}_{\alpha }$ such that, for each $\alpha ,$ the restriction of $F$ to $X_{\alpha }$ is a locally constant sheaf of finite dimensional vector spaces over $\overline{{\mathbb{Q}}_l}\text{.}$ A complex $A\in K\left(X\right)$ is $\overline{{\mathbb{Q}}_l}\text{-constructible}$ if ${\mathscr{H}}^i\left(A\right)$ is $\overline{{\mathbb{Q}}_l}\text{-constructible}$ for all $i\text{.}$

Definition of the category $D_c^b\left(X\right)$

Let $X$ be a variety. Let $A$ and $B$ be complexes of sheaves on $X\text{.}$ Define an equivalence relation on diagrams

$$A\stackrel{\sim }{⟵}C⟶B$$

in $K\left(X\right)$ which have $A$ and $B$ as end points by saying that the diagram $A\stackrel{\sim }{⟵}C⟶B,$ is equivalent to the diagram $A\stackrel{\sim }{⟵}{C}^{\prime }⟶B,$ if there exists a commutative diagram

$$\begin{array}{ccccc}& & C& & \\ & ⟶ & \uparrow & ⟶ \\ A& \stackrel{\sim }{⟶}& D& ⟶& B\\ & ⟶ & \downarrow & ⟶ \\ & & C^{\prime }& & \end{array}$$

The notation $C\stackrel{\sim }{⟵}A$ denotes that the map is a quasi-isomorphism. The bounded derived category of $\overline{{\mathbb{Q}}_l}\text{-constructible}$ sheaves on $X$ is the category $D_c^b\left(X\right)$ given by

1. Objects: Bounded, $\overline{{\mathbb{Q}}_l}\text{-constructible}$ complexes of sheaves on $X\text{.}$
2. Morphisms: A morphism from $A$ to $B$ is an equivalence class of diagrams $A\stackrel{\sim }{⟵}C⟶B\text{.}$

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 $A\stackrel{\sim }{⟵}B$ is a morphism from $A$ to $B\text{.}$

Given two morphisms $A\stackrel{\sim }{⟵}D⟶B$ and $B\stackrel{\sim }{⟵}E⟶C$ in $D_c^b\left(X\right)$ one can show that there always exists a commutative diagram

$$\begin{array}{ccccc}& & & & F\\ & & & ⟶ & & ⟶ \\ & & D& & & & E\\ & ⟶ & & ⟶ & & ⟶ & & ⟶ \\ A& & & & B& & & & C\end{array}$$

and one defines the composition of the two morphisms $A\stackrel{\sim }{⟵}D⟶B$ and $B\stackrel{\sim }{⟵}E⟶C$ to be the morphism defined by the diagram $A\stackrel{\sim }{⟵}F⟶C\text{.}$

Functors

The direct image with compact support functor $f_{!}$

A map $g:X\to Y$ between locally compact algebraic varieties is compact if the inverse image of every compact subset of $Y$ is a compact subset of $X\text{.}$

Let $f:X\to Y$ be a morphism of locally compact algebraic varieties. Let $F$ be a sheaf on $X\text{.}$ The support, $\text{supp}s,$ of a section $s$ of $F$ on an open set $V$ is the complement in $V$ of the union of open sets $U\subseteq V$ such that $s{\mid }_U=0\text{.}$

The direct image with compact support sheaf $f_{!}F,$ is the sheaf on $Y$ defined by setting

$$\Gamma (U;f_{!}F)=\{s\in \Gamma (f^{-1}\left(U\right);F)\mid f:\text{supp}s\to U\text{is compact}\},$$

for every open set $U$ in $Y\text{.}$ (For a sheaf $F$ on $x$ and an open set $U$ in $X,$ $\Gamma (U;F)=F\left(U\right)\text{.)}$ This defines a functor $f_{!}:\text{Sh}\left(X\right)\to \text{Sh}\left(Y\right),$ where $\text{Sh}\left(X\right)$ denotes the category of sheaves on $X\text{.}$

Let $f:X\to Y$ be a morphism of locally compact algebraic varieties. The direct image with compact support functor $f_{!}:D_c^b\left(X\right)\to D_c^b\left(Y\right)$ is given by

$$f_{!}=R{f}_{!},$$

so that $f_{!}$ is the right derived functor of the functor $f_{!}:\text{Sh}\left(X\right)\to \text{Sh}\left(Y\right)\text{.}$

The inverse image functor $f^{*}$

Let $f:X\to Y$ be a morphism of algebraic varieties. Let $F$ be a sheaf on $Y\text{.}$ The inverse image sheaf $f^{*}F$ is the sheaf on $X$ associated to the presheaf

$$V⟼\underset{U\supseteq f\left(V\right)}{\text{lim}}F\left(U\right),\text{for all}V\text{open in}X,$$

where the limit is over all open sets $U$ in $Y$ which contain $f\left(V\right)\text{.}$ This defines a functor $f^{*}:\text{Sh}\left(Y\right)\to \text{Sh}\left(X\right)\text{.}$

The functor $f_{♭}$

Let $f:X\to Y$ be a morphism of algebraic varieties. Let $A\in D_c^b\left(X\right)\text{.}$ Then $f_{♭}A$ is the unique (up to isomorphism) complex on $Y$ such that

$$A\cong f^{*}\left(f_{♭}A\right)\text{.}$$

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

The shift functor $\left[n\right]$

Let $A$ be a complex of sheaves on $X\text{.}$ For each integer $n$ define a new complex $A\left[n\right],$ with differentials $d{\left[n\right]}_i,$ by

$${\left(A\left[n\right]\right)}^i=A^{n+i},\text{and}{\left(d\left[n\right]\right)}_i={(-1)}^n{d}_{n+i}\text{.}$$

The shift functor is the functor

$$\begin{array}{ccc}{D}_c^b\left(X\right)& \stackrel{\left[n\right]}{⟶}& D_c^b\left(X\right)\\ A& ⟶& A\left[n\right]\text{.}\end{array}$$

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, $\text{supp}F,$ of a sheaf $F$ on $X$ is the complement of the union of open sets $U\subseteq X$ such that $F{\mid }_U=0\text{.}$

A complex $A\in D_c^b\left(X\right)$ is a perverse sheaf if

1. $\text{dim supp}{\mathscr{H}}^i\left(A\right)=0$ for $i\ge 0$ and $\text{dim supp}{\mathscr{H}}^i\left(A\right)\le -i$ for $i<0,$ and
2. $\text{dim supp}{\mathscr{H}}^i\left(D\left(A\right)\right)=0$ for $i\ge 0$ and $\text{dim supp}{\mathscr{H}}^i\left(D\left(A\right)\right)\le -i$ for $i<0,$

where $D\left(A\right)$ is the Verdier dual of $A\text{.}$

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 $D_c^b\left(X\right)$ whose objects are perverse sheaves on $X$ is an abelian category.

Intersection cohomology complexes

Let $Y\subseteq X$ be a smooth locally closed subvariety of complex dimensional $d>0$ and let $ℒ$ be a locally constant sheaf on $Y\text{.}$ There is a unique complex $IC(Y,ℒ)$ in $D_c^b\left(X\right)$ such that

1. ${\mathscr{H}}^i\left(IC(Y,ℒ)\right)=0,$ if $i<-d,$
2. ${\mathscr{H}}^{-d}\left(IC(Y,ℒ)\right){\mid }_Y=ℒ,$
3. $\text{dim supp}{\mathscr{H}}^i\left(IC(Y,ℒ)\right)\le -i,$ if $i>-d,$
4. $\text{dim supp}{\mathscr{H}}^i\left(D\left(IC(Y,ℒ)\right)\right)\le -i,$ if $i>-d,$

The complexes $IC(Y,ℒ)$ 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 $IC(Y,ℒ)$ as $ℒ$ runs through the irreducible locally constant sheaves on various smooth locally closed subvarieties $Y\subseteq X\text{.}$

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.

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