These are my live-TeXed notes for the course Mathematics 262y: *Perverse Sheaves in Representation Theory* taught by Carl Mautner at Harvard (Fall 2011). I eventually put some effort and editted the part before the Verdier duality during the winter. Please let me know if you notice any errors or have any comments!

##
The Lefschetz hyperplane theorem

(Lefschetz Hyperplane Theorem)
Let

be a projective variety and

be the intersection of

with a hyperplane

such that

is smooth, then the map

induced by the inclusion

is an isomorphism for

and an injection for

.

Let

be a hypersurface of degree

. Then

for some hyperplane

, where

is the Veronese embedding. Therefore by the Lefschetz hyperplane thoerem, for

, we have

Moreover, when

is smooth, we have the same

for

by Poincare duality. If we replace

by a field, this holds except the middle degree

.

If is smooth, by Poincare duality we also have a *Gysin homomorphism* . The composition is given by , where is the first Chern class of the hyperplane section.

(Hard Lefschetz)
Let

be a smooth projective variety of dimension

. Then for any embedding

and

, the map

is an isomorphism.

The Hard Lefschetz can be generalized to any Kähler manifold. Let be a complex manifold. One can show that any can be endowed wit a Hermitian metric on . Write , where is a Riemann metric and is an anti-symmetric 2-form.

A

*Kähler manifold* is a complex manifold with a Hermitian metric

where

.

Any complex smooth projective variety is Kähler. The idea is to pull back the Hermitian metric from

.

(Kahler version)
Let

be a Kähler manifold of dimension

with Kähler class

. Then for

,

is an isomorphism.

##
Hodge theory on Riemannian manifolds

Let be a real vector space of dimension . Let be an inner product. Let be the exterior algebra of . Then there is an induced inner product on such that for .

An

*orientation* of

is a choice of a vector

of length 1. Define an operator

such that

is characterized by

. One can check that

.

Let be a Riemannian manifold. Then is an inner product on , which extends to a smooth inner product on . Let be the smooth -forms.

An

*orientation* on

is a choice of a top form on

with norm 1 in each fiber.

If

is oriented, we define the

*Hodge star operator* using the above local construction pointwise.

For

, we define the inner product on

by

Define

by

. Using Stokes' theorem, one can show that

.

Define the

*Laplacian* by

. The kernel

of

is called the space of

*harmonic forms*.

is harmonic if and only if

and

.

A direct check using definition.

(Hodge)
Let

be a compact oriented Riemannian manifold. Then there is an isomorphism (depending on

)

.

The proof of the Hodge theorem involves the analysis of elliptic operators in order to construct *Green's operator*, which we shall not get into here.

(Poincare Duality with real coefficients)
The pairing

is non-degenerated.

Let

be a harmonic form, then

is also harmonic by Proposition

1 and

So the pairing is non-degenerated.

¡õ
##
Complex manifolds and the -Hodge Theorem

Let

be a complex manifold of dimension

. We denote

the

*real tangent bundle* (think of

as real manifold). It is a real vector bundle of rank

with

. The

*complex tangent bundle* is defined to be

, namely

.

A map induces the tangent map . If the is holomorphic, then the tangent map (or, its Jacobian matrix) has very restricted form, since the subspaces and are preserved under holomorphic change of variables.

Define the

*holomorphic tangent bundle* to be the holomorphic subbundle of

with

. Similarly define

be the

*anti-holomorphic tangent* bundle with

.

We have -isomorphisms between and , and also .

Using the dual basis

, we have similar notions of

*real, complex, holomorphic and anti-holomorphic cotangent bundles*.

The space of

*complex-valued smooth -forms* is defined to be

.

For , nonnegative integers, we denote . Then

A

*-form* is an element of

, which can be locally written as

.

We have . Let be the projection. Define the *Dolbeault operators* by and similarly . Then and . We denote the corresponding *Dolbeault cohomology groups* of and by and .

Let be a Hermitian form, then is a positive definite quadratic form. Let be a complex manifold with the Hermitian metric , then we can define and on using . We can extend them -linearly to . Then and is a linear isometry. We can also extend the pairing on to by

Define

and

. Again using Stokes' theorem, one can show that

and

are adjoint pairs.

Define

and

. Define

. Define similarly for

.

(-Hodge Theorem)
Let

be a Hermitian manifold. Then there are isomorphisms of finite dimensional vector spaces

##
The Hard Lefschetz theorem on Kähler manifolds

A Hermitian manifold

is

*Kähler* if the real 2-form

is closed, i.e.,

.

Let

be a Kähler manifold. Define the

*Lefschetz operator* by

. As

, this induces a map

. Define

to be the adjoint to

. We have

since

on

-forms.

The actions of

and

form an action of

on

. This implies

, hence

.

By Corollary 2, to prove the Kähler version of the Hard Lefschetz Theorem, it suffices to show that sends harmonic forms to harmonic forms.

(Kahler identities)
- , .
- .
- .

For (a), see Griffiths-Harris. For (b), use (a). For (c), using the definition of

and (b), it suffices to check that

, which can be shown using (a).

¡õ
.

.

commutes with

.

As commutes with , we now know that sends harmonic forms to harmonic forms, which implies the Hard Lefschetz Theorem.

##
The Lefschetz decomposition and Hodge-Riemann bilinear relations

For

, we define

to be

, called the space of

*primitive -forms*. We define

to be

, called the space of

*primitive cohomology classes*.

The Hard Lefschetz Theorem then has the following easy consequence.

(Lefschetz Decomposition)
Any

can be written uniquely as a sum

, where

. For

, we have the

*Lefschetz decomposition*
We omit the proof of the above useful fact.

Let be the Poincare duality pairing. Define the *intersection form* on using , Then is symmetric when is even and skew-symmetric if is odd. Define the Hermitian form on by

g

The Lefschetz decomposition on

is an orthogonal decomposition, i.e.,

for

.

as

.

¡õ
(Hodge-Riemann bilinear relation)
The decomposition

is orthogonal with respect to

. Moreover,

is positive definite on

.

is a

-form in

. It is non-zero only when

and

, hence

, which implies the orthogonality. Let

, then there exists

primitive and harmonic with

. So

is also primitive and harmonic. Then by Lemma

2,

The positive definiteness then follows.

¡õ
##
Cohomology of sheaves and categories

We construct a sequence of vector bundles which is an injective ?? resolution of the trivial bundle. The global sections form a complex and its cohomology is the de Rham cohomology. More generally, we would like to replace the vector bundles by any sheaves. In abstract language, we would like to define a new category of sheaves such that

- Any object in should be identified with all its resolutions.
- Functors should only be applied to special representatives in the isomorphism class of an object to obtain its cohomology.

A morphism of

of complexes in an abelian category is called a

*quasi-isomorphism* if the induced morphism on the cohomology

is an isomorphism.

Let

be a resolution. Then the morphism

is an quasi-isomorphism.

Let be a category and be a class of morphisms in . We can construct a functor such that for any , is an isomorphism, where has the same object as with morphisms in formally inverted. However, this construction loses the additive structure. To solve this problem, we shall only do this construction for a localizing system.

Unfortunately, the class of quasi-isomorphisms does not form a localizing system for . In order to construct the derived category, we need to pass to the homotopy category . It turns out that the quasi-isomorphisms form a localizing system for and the category is the desired derived category .

##
Aside on spectral sequences

Let

. A

*decreasing filtration* is a sequence

Let

be a complex, a

*decreasing filtration* is a double sequence

such that

and

. The filtration on

induces a filtration on

given by

.

Let

be a double complex and

be the total complex. Define

. The associated spectral sequence for this filtration is

and

.

##
Ordinary cohomology = Sheaf cohomology of the constant sheaf

If

is a contractible space, then

.

Cf. Corollary 2.7.7 (iii) of Kashiwara M., Schapira P, *Sheaves on manifolds*.
¡õ

Let

be a sheaf on

,

be an open cover of

, such that for any

,

,

. Then

.

To define the Cech cohomology

, we form

with

So we get a complex of sheaves

on

by restrictions. The sequence

is a resolution, hence

is quasi-isomorphic to

. Let

be a flabby resolution of

. Then we can construct with the Cech complex of the

's a double complex

. Consider the spectral sequence from the filtration

of the global sections. As

is flabby and the columns are resolutions, we get

And

Using the other filtration we know that

.

¡õ
If

is a CW-complex, then

.

Choose a cover of

such that

is contractible for any

and apply the previous proposition.

¡õ
Let

be a continuous map, then

is the sheafification of the presheaf

Let

be an injective resolution. Then

is the sheafification of

.

¡õ
So gives a good notion of cohomology in *families*. For example, to compute the cohomology of , we can compute for any map .

##
Degeneracy of the Leray spectral sequences

Let be abelian categories and be a left exact functor. Suppose has a class of -acyclic objects.

For any

, there exists a spectral sequence

.

More generally, let , where , are two left exact functors. Suppose has a class of -acyclic objects and has a class of -acyclic objects such that is -acyclic.

(Grothendieck spectral sequence)
For any

, there exists a spectral sequence

.

Let be a smooth fiber bundle with smooth compact fibers . Then the Grothendieck spectral sequence associated to applied to the constant sheaf becomes the classical *Leray spectral sequence* where is the local system on with fiber .

For the Hopf fibration

,

and

are constant sheaves as

.

A

*family of projective manifolds* is a proper, holomorphic submersion of smooth varieties

factoring through

with fibers smooth projective varieties.

Deligne proved the degeneracy of the Leray spectral sequence for a family of projective manifolds.

(Version II implies Version I)
Let

. The Leray spectral sequence is the spectral sequence

applied to

and it is degenerate if

is concentrated in a single degree. If

, then the Leray sequence is the direct sum of the spectral sequence of

. Hence the Leray spectral sequence degenerates.

¡õ
(Version II)
Suppose

. Let

be the first Chern class of the hyperplane section. By the previous remark, we have

and induces

. On each fiber,

is an isomorphism by the Hard Lefschetz. Therefore

is an isomorphism on each stalk, thus

is an isomorphism. Now applying the following Key Lemma to

,

and

the Lefschetz operator, we know that

.

¡õ
(Key Lemma)
Let

be an abelian category. Let

and

such that the induced maps

are isomorphisms. Then

.

(van den Bergh)
By induction downward on

, where

is an integer such that

for any

. Then

is the isomorphism induced by

(by taking

). Using

We get a map

and

. And

. Using properties of triangles, we get

. To finish the proof, we check that

whenever

.

¡õ
Next part of this course is to define more operations on sheaves and establish the Poincare duality of sheaves for singular varieties.

##
Operations on sheaves

Let be a commutative ring and . We define be the sheaf . Then . Because is a left exact functor, we obtain a right derived functor .

Similarly, we define by . Then . Because is right exact, we obtain a left derived functor . is called *flat* if is exact. Note that is flat if and only if is a flat -module for any . Flat sheaves form an acyclic class for .

.

.

If

is locally free and

injective, then

is injective. So

The general case follows by taking a locally free resolution of

and an injective resolution of

.

¡õ
Let

be a map of topological spaces. For any open subset

, let

. Then

forms a subsheaf of

, called the

*direct image with compact support*. Note that

is a left exact functor.

Define

. Then

for the morphism

.

Let

, where

locally compact. Then for any

,

is an isomorphism.

For any

, there exists

an open neighborhood of

and

such that

is proper. It follows that

has compact support

. We define

. One can check that

is injective. For surjectivity, we use the following lemma. Hence

is an isomorphism.

¡õ
If

is Hausdorff (resp. paracompact),

is compact (resp. closed). Then

is an isomorphism (i.e. we do not need to sheafify).

Consider

. Then

as the embedding

is never proper.

A sheaf

is

*soft* if for any

compact, the restriction map

is surjective.

is soft if and only if for any closed subset

, the restriction map

is surjective.

If

is soft, then for any locally closed embedding

,

is also soft.

For any

closed, we have the surjection

by the softness of

.

¡õ
If

is exact with

soft. Then

is also exact. In particular,

is exact.

By Corollary

8,

is soft for any

. Since it is enough to check the exactness on stalks, we reduce to the exactness of

by Proposition

5. By the left exactness of

, we only need to check the surjectivity of

. Let

. Choose a compact open subset

. Replace

by

and

by

, we may assume that

is compact. Giving

is the same as giving a finite compact cover

of

and

such that

. One can check

for some

. By the softness of

, we get a global section

maps to

. Replace

by

, then

. Now an induction shows that

can be glued to be a section of

.

¡õ
If

is exact and

are soft, then

is soft too.

For any

closed, we have the following diagram

Hence

is soft.

¡õ
The above two propositions together imply the following theorem.

Soft sheaves form an acyclic class for

.

In fact more is true:

If

is locally compact and countable at

(its one-point compactification is Hausdorff). Then soft sheaves form an acyclic class for

.

The de Rham resolution is a soft resolution, hence

.

By Remark 13, we know there exists enough soft sheaves. So we have the right derived functor . In particular, we have the derived functor . Define the *cohomology with compact support* . Note that .

(Proper Base change)
If we have a Cartesian diagram

Then there exists a canonical isomorphism

.

There exists a canonical map

. By adjunction of

, have

, which corresponds to

. This induces an isomorphism.

¡õ
(Projection formula)
There exists a natural map

. It is an isomorphism if

is flat.

- .
- .

(Stalks of )
is naturally isomorphic to the sheafification of the presheaf

. Denote

. As

is exact, we know that

. For example, let

and

, then

.

(Stalks of )
Using the base change

we know that

.

We have seen in the exercise that if is a closed embedding, then has a right adjoint , where is functor of taking the sheaf of sections with support inside . On the contrary, suppose , then does not admit a right adjoint. Does admit a right adjoint in the bounded below derived category? Or more generally, does admit a right adjoint functor for a continuous map between locally compact spaces? The answer is YES.

There exists a right adjoint

to

. We call

the

*exceptional inverse image functor*. (See the next section for a brief discussion of

.)

We define the

*dualizing sheaf* , where

.

For a oriented manifold, . In particular, by adjunction we have On the other hand, So in this way we recover the Poincare duality. (In general, for unoriented manifolds, the dualizing sheaf is the *orientation sheaf* shifted by the dimension.)

More generally, for any . We have

We define the

*dualizing functor*
Given a "nice" singular space

, can we associate to

some canonical object in

that is self-dual, i.e.,

? If it is the case, then we will obtain a desired analog of the Poincare duality for singular spaces.

##
Verdier duality

The original proof existence of is due to Verdier, using that we already know that exists for a locally closed embedding and then gluing them together. It is difficult since the derived category does not have good gluing property. Instead of Verdier's approach, we will give a proof due to A. Neeman.

For any abelian category

, the categories

and

are triangulated categories.

(Alonso-Jeremias-Souto, Neeman)
The unbounded derived theory of a Grothendieck abelian category is well generated. (A *Grothendieck abelian category* is an abelian category with generators such that small colimits and filtered colimits are exact.)

(Brown representability)
Let

be two triangulated categories and

be well generated and with arbitrary coproducts. Then a functor

admits a right adjoint if and only if

commutes with coproducts.

(Spaltenstein)
For any

locally compact spaces,

is defined on

*all* of

.

commutes with arbitrary direct sums.

By the lemma and the Brown representability, we conclude that has a right adjoint functor . To get a bounded functor , we need further boundedness of .

The

*dimension with compact support* for

locally compact is the smallest

such that

for any

and any

.

- .
- If is closed, then .
- is local, namely if for any , there exists a neighborhood of such that , then . In particular, for any -dimensional manifold.
- For and , then for any and . Moreover, .

Now assume that . Let , then by adjunction Suppose , then we know that . So for , we have , hence . It follows that , hence the adjunction can be defined on . This adjoint pair is called the *(global) Verdier duality*.

Let . Then there is a canonical map Deriving this, for any , we get Replacing by for some , we obtain

(Local Verdier duality)
The map

is an isomorphism.

We check that

is an isomorphism on each open

:

by global Verdier duality, the right-hand-side is isomorphic to

The proposition follows.

¡õ
We also have the following similar useful identity and base change.

If

and assume that

has fibers of finite dimensions (hence so does

), then

.

The idea is to use the adjunction

.

¡õ
##
Contraction of curves on complex surfaces

Throughout this section, we assume that the coefficient ring is a field. Let be a quotient map such that and , namely a contraction of the union of curves on a complex surface to a single point .

(Grauert)
is holomorphic if and only if the intersection matrix

is negative definite.

The contraction

has

, hence is not holomorphic.

Let

be a line bundle over

such that

, then the contraction of its zero-section on the total space of

has negative definite intersection pairing

, hence is holomorphic.

Consider the blowup

, the contraction of

(

is the strict transform of

.

is the exceptional divisor) has

, hence is not holomorphic.

Consider the blowup

has

, hence is holomorphic.

Now restrict to one of our four examples. Let and , . The following is a fact from basic algebraic topology.

(Lefschetz duality)
.

Since retracts onto , we have . Let be the cycle class map given by and and . There is a long exact sequence associated to , by the Lefschetz duality we have the following identification:

Note that , so is an isomorphism if and only if it is surjective if and only it is injective, which is also equivalent to say that is an isomorphism, or , or , or is an isomorphism, if and only if is non-degenerate (e.g., when and is holomorphic.)

What does this mean in

?

Consider . Then

Consider the truncation triangle

Does this split? Namely, does there exist

such that

is an isomorphism?

By the exact sequence of sheaves We know that for any , we have a distinguished triangle Taking its cohomology, we have a long exact sequence Hence is actually the relative cohomology.

has support on a single point . By applying to the above distinguished triangle,we get Hence must factor through . Taking , we get . By base change, . We conclude that the sequence splits if and only if is an isomorphism.

When does the complex

split?

If the intersection pairing

is invertible, then

. (Note

and

.)

When

is invertible, we want a splitting

. Since

is supported on

. Then

exists if and only if there exists a map

, which live in

. Thus it is equivalent to giving a map

such that

is the identity map, which is equivalent to saying that

is an isomorphism,

is invertible.

¡õ
Truncating the adjunction map , we get , which is an isomorphism in . We can check this on locally, as and , which an isomorphism if and only if is an isomorphism.

We denote

,

,

. More generally, for any smooth

, we denote

. Then if

is holomorphic and

, then

.

In the above non-holomorphic examples,

does not split. However, one can always split off a skyscraper sheaf of rank

.

Here is another approach. Consider the adjunction map .

When does it split?

Again, truncating the adjunction map , we get as for . Note that and for . We obtain that
We get given by . Applying to the triangle, we get which sends to . Then lifts if and only if , if and only if is an isomorphism.

Thus exists if and only is invertible. Similarly form the triangle , we know that is an isomorphism. Therefore exists (and is unique) if and only if is an isomorphism and (in this case ).

##
Borel-Moore homology and dualizing functor

Last time we studied the pushforward of constant sheaves and how they decompose. Now let us step back to duality.

We have seen that cohomology can be naturally expressed in terms of sheaves.

What about homology?

Let be the chain complex of possibly infinite singular (simplicial) chains on together with the usual differential such that for any compact , there exists at most finitely many such that with . Its homology is called the *Borel-Moore homology*.

- In , a ray is a 1-chain. Its boundary is a single point and itself is a boundary. Hence and . Also, .
- Consider a three rays in plane branched at one point. Then its Borel-Moore and .

Let

.

is in fact a complex of flabby sheaves, so

.

The Poincare duality for smooth oriented manifolds can be also stated as .

The universal coefficients theorem says that By the Poincare duality, we obtain where can be also identified as .

We want a general notion of a *dual complex* for any such that

Given a complex

, we let

be the complex of presheaves

and define

to be its sheafification. In fact,

.

Let , then the Borel-Moore homology is the dual of its cohomology with compact support. Hence . It follows from the calculation of basic Borel-Moore homology calculation that

Let

be a smooth manifold of dimension

. Then

, where

is the orientation sheaf.

We would like to say that gives us a dualizing functor , with . Unfortunately, is a bit too wild for this to be true. The first problem is that the image of the functor may not lie in . This is not a major problem (e.g. for , which works out). The second problem is that there may exist bad sheaves on nice spaces. For example, let and be locally constant sheaf on and consider the sheaf on . We would like to eliminate those problems.

An

*analytic space* is a subset of an analytic manifold of

cut out by analytic functions. A

*subanalytic space* is one cut out by analytic equalities and inequalities.

A sheaf

on a subanalytic space

is called

*constructible* if there exists

a subanalytic locally finite stratification (i.e.,

is a subanalytic subspace which is also a manifold) such that

is a local system, i.e., a locally constant sheaf.

Let

. We say

is

*(cohomologically) constructible* if

is constructible.

The

*constructible derived category* is defined to be the full subcategory of

consisting of cohomologically constructible complexes.

- Let be an analytic map, then all preserve .
- In , is a dualizing functor, i.e., .

Given . We have as , hence . Similarly, we have

##
Stratification

Suppose we have a locally finite covering of smooth subanalytic subsets , we say that it is a (Whitney) *stratification* if it satisfies and the Whitney conditions A and B. The condition A says that is contained the limit of the tangent spaces for any and in the strata. It follows under these conditions that there exists a neighborhood of such that is strata-preserving homeomorphic to , where is the *link* of . Let be a transverse slice to the strata containing , then . The following are the basic facts about stratification.

- Any locally finite covering by subanalytic subsets can be refined to a stratification.
- Any algebraic variety admits a stratification by locally closed subvarieties.
- Any map of varieties can be stratified (namely, there exists stratifications of and such that the preiamge of a strata is a union of strata such that is a submersion and is locally constant over .)

We have seen that , When is smooth and oriented, we have , in fact for any smooth morphism . Let , we obtain the Poincare duality This can be interpreted as the statement that is *self dual*.

An analog of Poincare duality on stratified space of even real dimension should then be such that and an open smooth (strata) such that .

We have seen that for a smooth projective map , and by the Hard Lefschetz. Also, we have seen that for a contraction of curves , . As is self dual, is self dual (as is proper) and is self dual, we know that is self dual.

In general for proper, we cannot hope , but instead we would hope that , where is the form of .

##
Poincare duality for singular spaces

For singular, usually . In order to obtain the Poincare duality for singular spaces, we need to find some such that and for some open such that . This is the goal for today.

We will go by induction. Let , where is smooth closed (but is not necessarily smooth). Let is open and . Assume there exists a stratification of such that is smooth closed stratum and is constructible with respect to the stratification.

An

*extension* of

is a pair

, where

and

an isomorphism.

Fix

, then there exists a natual bijection

given by sending

to

.

The sheaf

correpsonds to

. Namley,

.

The sheaf correpsonds to .

From , we get corresponding to , hence we get . So for to be self-dual, we need and . So we want to find such that . The (only) way to find a splitting is by trucation: As is locally constant on , we know that Decompose as and , where is contractible and open. Then , where 's are all constant sheaves.

What is the "costalk", namely

of a constant sheaf on a smooth

of dimension

?

As the dualizing sheaf of

is

, which is also equal to

. Also

. Hence

.

Therefore we . Applying to , we obtain that Hence So So we should take , namely .

To summarize, the above procudure works for even and . Starting with a self-dual sheaf , we get sheaf given by the extension corresponding to the distinguished triangle

In fact, we will see that defines a functor .

Consider the long exact sequence of

, we know (a) is equivalent to (b) and the uniqueness. From the triangulated category axiom, we know that (b) is equivalent to (d). A dual argument implies that (c) is equivalent to (d).

¡õ
is a functor.

Since

and

as

and

. By Corollary

11, we know

is functorial.

¡õ
has a nontrivial summand with support in

ifa nd only if

can be expresseed as

, where

,

and

.

If

, then

is the same as

and

.

For the other direction, we know is a direct summand of as the map is the identity.
¡õ

If

for any

, then

has no summands with support in

.

If not, then

. But

is nonzero for some

.

¡õ
has no summands with support in

.

Since

for any

and

for any

. Hence

for any

.

¡õ
Let

be stratified by decreasing dimensions (so

is open in

). Let

is locally constant on

. Define the

*intersection cohomology complex* , where

. Write

.

We saw that . Also, is indecomposable if and only if is indecomposable.

Define the

*intersection cohomology* and

.

As is self-dual, we get the Poincare-Verdier-Goresky-MacPherson duality

Now many results can be extended to singular varieties using intesection cohomology.

The Lefschetz hyperplane thoerem is true for any projective variety with

replaced by

.

The Hard Lefschetz theorem is true for any projective variety with

replaced by

.

Recall that for a projective smooth morphism , we have , where is a semisimple local system by Deligne's theorem. Moreover, .

(Decomposition theorem, BBD)
If

is proper and

is smooth, then

Our next goal is to "filter" in such a way that the -th "associated graded piece" is . Namely, the following relative Hard Lefschetz holds: and .

##
-structures

Let be a triangulated category. Let and be full categories of . Let and .

is called the

*heart* or

*core* of the

-structure.

Let

be an abelian categroy, then

and

gives a

-structure on

with heart equivalent to

.

- There exists a functor which is right adjoint to the inclusion . Similarly for a functor .
- There exists a unique such that is a distinguished triangle.

Define

and

. We want to show that given

, there exists a canonical map

. Applying

to

, we get

. Since

and

, we know

, hence

is a functor.

The same argument shows that . Since , we know that there exists a unique .
¡õ

- If , then . Similarly for .
- Let . Then if and only if . Similarly for .
- If is a distinguished triangle in and , then .

For (a), use the adjunction from the last proposition. For (b), use (a). For (c),

and the long exact sequence implies that

.

¡õ
- If , then . Similarly for .
- If , then .
- More generally, .

Let

be the heart. We define

, where

and

. So

if and only if

for every

.

The heart

is an abelian categroy.

Note that for any

, the distinguished triangle

shows that

. Thus

is additive. For

in

, the distinguished triangle

shows that

. We claim that

and

. This claim can be checked using

and

. It remains to check that

. Define

such that

is a distinguished triangle. Then

and

. By completing into a tetrahedron, we get an triangle

Hence

and

. Hence

and

.

¡õ
The functor

is cohomological, i.e., for every triangle

in

, we obtain a long exact sequence of

By rotation, it suffices to show that

is exact.

- Assume , we shall show that is exact. In this case, we have and for any . Applying , we know the short exact sequence.
- Assume that only . We shall show the short exact sequence . By applying for any , we know . By completing into a tetrahedron, we have the triangle . By the first step, we know is exact.
- Run the same argument, we know that if , then is exact.
- Let be arbitrary. Let such that is a distinguished triangle. Then is exact by the third step. We also have the triangle , hence is exact by the second step. It follows that is exact.
¡õ

##
Perverse -structures

Let

. We say

satisfies the

*support condition* if

for any

. We define

to be the full subcategory of objects satisfying the support condition.

We say

satisfies the

*cosupport condition* if

. We define

to be the full subcategory of objects satisfying the cosupport condition.

is a

-structure on

(called the

*perverse -structure*).

We define

to be the categroy of perverse sheaves in the heart of the perverse

-structure.

Why do we define the perverse

-structure in this way?

Here is another approach. Let be a fixed stratification and be the complexes constructible with respect with this stratification. We would like to construct a self-dual -structure on individually using descending induction on the strata.

On the top strata, we define to be the complex of sheaves on with local systems on . Let , for a local system on , we have , where . So is not self-dual, but instead is self-dual.

Now let and be the open and closed embeddings obtained from the strata. Suppose we already have a self-dual -structure on and also a self-dual -structure on given by , where . We want a self-dual -structure on such that and This is self-dual by definition, so we only need to check it is actually a -structure.

- Let and . Applying to the adjunction distinguished triangle of , we obtain Since and by construction, we know that .
- Since shifts commute with restrtion, we know that .
- For , we construct . We then construct such that and such that By completing the tetrahedron, we check that and . In fact, we have and , hence . Also, and , hence .
¡õ

It follows from the inductive construction that and This explains why we defined perverse -structures in such a way.

Notice that when is smooth, consists of complexes where are local systems. Then is th self-dual -structure on .

Let be the *degenerate* -structure on . By gluing with , we obtain a -structure on . Let be the corresponding truncating functor. Then is the right adjoint of the inclusion of objects such that and Dually, define by using the -structure glued from . We have

Let

and

. Then there exists a unique up to a unique isomorphism extension

of

such that

and

. Namely

Use Lemma

9 and notice that

.

¡õ
From and , we have a morphism of functors .

The image functor

is

.

Using the triangle for

, we have a short exact sequence

Similarly we have

It follows that

by identifying

with

.

¡õ
##
Simple objects

The simple objects in

are the

-sheaves.

From the strata and , we have and . Let be the essential image of in , in other words, the full subcategory with objects such that . Consider and the triangle , applying , we have a long exact sequence We know that is the maximal quotient of with support in . Similarly, is the maximal subobject of with support in .

The functor

factors through the Serre quotient categroy

. Moreover,

is an equivalence of categories.

is faithful: Let

and

. Let

be a lift of

. Since

, we know that

, so

, therefore

.

is essentially surjective and full: As .
¡õ

- For , is the unique extension of which has no nontrivial subquotients with support in .
- The simple objects of are
- for simple,
- for simple.

- Recall that is the maximal quotient ojbect of with support in and is the maximal suboject of with support in . So if has no subquotients with support in , then . Hence and , which is equivalent to . Now use the triangle .
- Any simple object in is either
- the image of a simple object in ,
- an extension of a simple object in which has no nontrivial subobjects with support in , i.e. for simple.
¡õ

##
Operations on perverse sheaves

Suppose and for any . Then Hence . By duality, we know that for , .

Let

and

be an adjoint pair of triangulated functors. Then

(

*right -exact*) if and only if

(

*left -exact*).

and

.

If is proper, then , hence . Recall that if is smooth of relative dimension , then . Hence and they take perverse sheaves to perverse sheaves (*-exact*). In particular, when is etale, we know that is -exact.

Recall that if is smooth and affine, then for any . The following is a generalization of this fact. The proof can be given by generalizing the original proof using Morse theory.

Let

be proper.

is locally closed subvarieties and

. We say

is

*semismall* (resp.

*small*) if

for any

(resp.

holds for any

).

Now suppose is semismall.

.

Let

be smooth. Then

.

has stalks

. So it is zero if

. So

as

. Namely

. One can similarly check that

.

¡õ
We showed that . Let . If, then , hence . In other words, the fibers over have dimension . is a local system on , hence if and only if for all . This explaines why we defined "semismall" in such a way.

##
Kazhdan-Lusztig conjecture

Fix a connected reductive algebraic group and . Let be the associated Weyl group.

The

*Hecke algebra* is defined to be

where

(Iwahori)
Let

,

. Then

. ??

The endomorphism of

given by

and

is an involution, denoted by

. We call

is self-dual if

.

The idea of the solution relates the with the geometry of . By the Bruhat decomposition, , where each . Write . Let be the constructible derived category with respect to the Bruhat stratification. For , by the decomposition theorem, we know that .

We define

by

.

, then

.

.

Then the Kazhdan-Lusztig conjecture follows from this geometric characterization.

<definition>
For

is called

-even (resp. odd) if

for all

odd (resp. even) (Equivalently,

for all

and

odd (resp. even).

is called

-parity if

.

Let

be a distinguished triangle in

. If

are

-even, then so is

. Moreover,

.

Applying

and using the long exact sequence in cohomology, we know that

and

.

¡õ
Let be the parabolic subgroup (e.g., block upper triangular matrices for ). Let .

(Push-Pull (Springer, Brylinski, MacPherson))
If

is

-parity, then

.

Now we will use the Push-Pull Lemma to prove the main theorem, namely .