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

,
![$$\eta^i=[\omega]^i\cup-: H^{n-i}(X;\mathbb{R})\rightarrow H^{n+i}(X;\mathbb{R})$$](./latex/latex2png-PerverseSheaves_143885913_.gif)
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
![$[\eta,\nu]=(l-n)\Id$](./latex/latex2png-PerverseSheaves_81803086_-5.gif)
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
![$[\alpha]=a$](./latex/latex2png-PerverseSheaves_82096938_-5.gif)
. 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
![$$\xymatrix{ 0\ar[r] & X \ar[r]\ar[d] & 0 \ar[r]\ar[d] &0 \ar[r]\ar[d] &\cdots\\
0 \ar[r] & I_0 \ar[r] & I_1 \ar[r] & I_2 \ar[r] &\cdots}$$](./latex/latex2png-PerverseSheaves_154030643_.gif)
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
![$$E_1^{p,q}=\xymatrix{0 \ar[r] & 0 \ar[r]& 0\ar[r] & \cdots\\ \Gamma(K^0)\ar[u]\ar[r] & \Gamma(K^1)\ar[u]\ar[r] & \Gamma(K^2) \ar[r]\ar[u] & \cdots }$$](./latex/latex2png-PerverseSheaves_241656174_.gif)
And
![$$E_2^{p,q}=\xymatrix{0 \ar[r] & 0 \ar[r]& 0\ar[r] & \cdots\\ R^0\Gamma(\mathcal{F}) \ar[u]\ar[r] & R^1\Gamma(\mathcal{F})\ar[u]\ar[r] & R^2\Gamma(\mathcal{F}) \ar[r]\ar[u] & \cdots }$$](./latex/latex2png-PerverseSheaves_221881566_.gif)
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
![$Rf_*\underline{\mathbb{C}}_X=\bigoplus_iR^if_*\underline{\mathbb{C}}_X[-i]$](./latex/latex2png-PerverseSheaves_59956974_-5.gif)
, then the Leray sequence is the direct sum of the spectral sequence of
![$R^if_*\underline{\mathbb{C}}_X[-i]$](./latex/latex2png-PerverseSheaves_23388151_-5.gif)
. 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
![$\eta: \underline{\mathbb{C}}_X\rightarrow\underline{\mathbb{C}}_X[ 2 ]$](./latex/latex2png-PerverseSheaves_210654761_-5.gif)
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

,
![$X=Rf_*\underline{\mathbb{C}}_X[ n ]$](./latex/latex2png-PerverseSheaves_14994549_-5.gif)
and

the Lefschetz operator, we know that
![$Rf_*\underline{\mathbb{C}}_X\cong\bigoplus_iR^if_*\underline{\mathbb{C}}_X[-i]$](./latex/latex2png-PerverseSheaves_141194806_-5.gif)
.
¡õ
(Key Lemma)
Let

be an abelian category. Let

and
![$\eta: X\rightarrow X[ 2 ]$](./latex/latex2png-PerverseSheaves_201720565_-5.gif)
such that the induced maps

are isomorphisms. Then
![$X\cong\bigoplus_iH^i(X)[-i]$](./latex/latex2png-PerverseSheaves_91409100_-5.gif)
.
(van den Bergh)
By induction downward on

, where

is an integer such that

for any

. Then
![$$\alpha: H^{-d}(X)\cong\tau_{\le0}(X[-d])\rightarrow X[-d]\xrightarrow{\eta^d}X[d]\rightarrow\tau_{\ge0}(X[d])=H^d(X)$$](./latex/latex2png-PerverseSheaves_186573479_.gif)
is the isomorphism induced by

(by taking

). Using
![$$\xymatrix{H^{-d}(X) \ar[r]^\alpha \ar[d]^i& H^d(X)\ar[d]_p\\ X[-d] \ar[r]_{\eta^d} & X[d],}$$](./latex/latex2png-PerverseSheaves_27251913_.gif)
We get a map
![$\Psi: X\rightarrow H^{-d}(X)[d]\oplus H^d(X)[-d]$](./latex/latex2png-PerverseSheaves_37574521_-5.gif)
and
![$\Phi: H^{-d}(X)[d]\oplus H^d(X)[-d]\rightarrow X$](./latex/latex2png-PerverseSheaves_177695416_-5.gif)
. And

. Using properties of triangles, we get
![$X=H^{-d}(X)[d]\oplus X'\oplus H^{d}(X)[-d]$](./latex/latex2png-PerverseSheaves_45314932_-5.gif)
. 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
![\begin{equation*}
\xymatrix{ \Gamma_c(X;\mathcal{F})\ar[r]\ar@{->>}[d] & \Gamma_c(X;\mathcal{F}'')\ar@{-->>}[d] \\ \Gamma_c(Z;\mathcal{F}) \ar@{->>}[r] & \Gamma_c(Z;\mathcal{F}'').}
\end{equation*}](./latex/latex2png-PerverseSheaves_115430328_.gif)
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
![$$\xymatrix{ X'\ar[d]_{g'} \ar[r]^{f'} & Y'\ar[d]^{g} \\ X \ar[r]^f& Y.}$$](./latex/latex2png-PerverseSheaves_85568631_.gif)
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
![$$\xymatrix{ f^{-1}(y) \ar[r]\ar[d]& X\ar[d]\\ \{y\}\ar[r] & Y,}$$](./latex/latex2png-PerverseSheaves_218191999_.gif)
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

:
![$$H^i(R\Gamma(U; Rf_*R\mathcal{H}om(\mathcal{F},f^!\mathcal{G})))\cong \Hom_{\mathcal{D}(f^{-1}(U)}(\mathcal{F}|_{f^{-1}(U)}, f^!\mathcal{G}[i]|_{f^{-1}(U)}),$$](./latex/latex2png-PerverseSheaves_94196197_.gif)
by global Verdier duality, the right-hand-side is isomorphic to
![$$\Hom_{\mathcal{D}(U)}(R(f|_U)_!\mathcal{F}|_U, \mathcal{G}[i]|_U)\cong H^i(R\Gamma(U; R\mathcal{H}om(Rf_!\mathcal{F},\mathcal{G}))).$$](./latex/latex2png-PerverseSheaves_66582413_.gif)
The proposition follows.
¡õ
We also have the following similar useful identity and base change.
If
![$$\xymatrix{ X'\ar[d]_{g'} \ar[r]^{f'} & Y'\ar[d]^{g} \\ X \ar[r]^f& Y.}$$](./latex/latex2png-PerverseSheaves_85568631_.gif)
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
![$[E_j\cdot E_k]$](./latex/latex2png-PerverseSheaves_193898993_-5.gif)
is negative definite.
The contraction

has
![$I=[0]$](./latex/latex2png-PerverseSheaves_241137311_-5.gif)
, 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
![$I=[c_1(\mathcal{L})]$](./latex/latex2png-PerverseSheaves_119241293_-5.gif)
, 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:
![$$\xymatrix@!C=.9cm{
H^1(X, U) \ar[r] \ar@{=}[d]& H^1(X)\ar[r] & H^1(U\ar[r]) & H^2(X, U)\ar[r]\ar@{=}[d] & H^2(X)\ar[r]\ar@{=}[d]& H^2(U) \ar[r] & H^3(X,U)\ar@{=}[d]\ar[r] & H^3(X) \ar@{=}[d]\\
H_3(E)=0 & & & H_2(E) \ar[ur]^{\mathrm{cl}_E} \ar[r] & H^2(E) & & H_1(E) & H^3(E)=0
}$$](./latex/latex2png-PerverseSheaves_85458236_.gif)
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 ![$$\tau_{\le 1} Rf_*\underline{k}_X\rightarrow Rf_*\underline{k}_X\rightarrow R^2f_*\underline{k}_X[-2]\rightarrow.$$](./latex/latex2png-PerverseSheaves_44722167_.gif)
Does this split? Namely, does there exist
![$\sigma: R^2f_*\underline{k}_X[-2]\rightarrow Rf_*\underline{k}_X$](./latex/latex2png-PerverseSheaves_32415126_-5.gif)
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
}_v)\oplus\tau_{\le1}Rf_*\underline{k}_X\cong \underline{H^2(E)}_v\oplus \tau_{\le1}Rj_*\underline{k}_U$](./latex/latex2png-PerverseSheaves_2691917_-8.gif)
. (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
![$IC_X=\underline{k}_X[2 ]$](./latex/latex2png-PerverseSheaves_264886932_-5.gif)
,

,
![$IC_Y=\tau_{\le1}(Rj_*\underline{k}_U[2 ])$](./latex/latex2png-PerverseSheaves_217936575_-6.gif)
. More generally, for any smooth

, we denote
![$IC_X=\underline{k}_X[\dim X]$](./latex/latex2png-PerverseSheaves_60682367_-5.gif)
. Then if

is holomorphic and

, then

.
In the above non-holomorphic examples,
![$R^2f_*[-2]$](./latex/latex2png-PerverseSheaves_167827506_-5.gif)
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
![$\omega_M\cong \mathrm{or}_M[n]$](./latex/latex2png-PerverseSheaves_162310577_-5.gif)
, 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
![$i^*\mathcal{F}\rightarrow i^*j_*\mathcal{F}_U\rightarrow i^!\mathcal{F}[1 ]$](./latex/latex2png-PerverseSheaves_205767865_-5.gif)
.
The sheaf

correpsonds to
![$i^*j_!\mathcal{F}_U=0\rightarrow i^*j_*\mathcal{F}_U\rightarrow i^!j_!\mathcal{F}_U[1 ]\rightarrow$](./latex/latex2png-PerverseSheaves_137961504_-5.gif)
. Namley,
![$i^*j_*\mathcal{F}_U\cong i^!j_!\mathcal{F}_U[1 ]$](./latex/latex2png-PerverseSheaves_129370685_-5.gif)
.
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
![$i_x^!a^!\underline{k}_\mathrm{pt}=i_x^!(\underline{k}_C[d])$](./latex/latex2png-PerverseSheaves_115393504_-7.gif)
. Also
![$i_x^*\underline{k}_C[d]=\underline{k}_\mathrm{pt}[d]$](./latex/latex2png-PerverseSheaves_84491503_-7.gif)
. Hence
![$i_x^!=i_x^*[-d]$](./latex/latex2png-PerverseSheaves_104415591_-5.gif)
.
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
![$j_{!*}\mathcal{F}=\mathrm{Cone}(i_*A[-1]\rightarrow j_!\mathcal{F}_U)$](./latex/latex2png-PerverseSheaves_186310967_-5.gif)
and
![$$\Hom^{-1}(i_*A[-1],j_{!*}\mathcal{F}_U)=\Hom(A[-1],i^!j_{!*}\mathcal{F}_U[-1])=\Hom(A, B[-1])=0$$](./latex/latex2png-PerverseSheaves_216669975_.gif)
as

and
![$B[-1]\in\mathcal{D}^{\ge-d/2+1}$](./latex/latex2png-PerverseSheaves_27241203_-5.gif)
. 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
![$H^i(\tau_{\ge-d}i^*j_*\mathcal{F}_U)[-1]=0$](./latex/latex2png-PerverseSheaves_80520451_-6.gif)
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 ![$IC(X; \mathcal{L})=(j_n)_{!*}\cdots (j_0)_{!*}\mathcal{L}[\dim X/2]$](./latex/latex2png-PerverseSheaves_26525307_-5.gif)
, 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
![$\Hom(X_0, Y_1[-1])\rightarrow \Hom(X_0,Y_0)\rightarrow\Hom(X_0, Y)\rightarrow\Hom(X_0, Y_1)$](./latex/latex2png-PerverseSheaves_99791975_-5.gif)
. Since
![$\Hom(X_0, Y_1[-1])=0$](./latex/latex2png-PerverseSheaves_262979685_-5.gif)
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
![$\mathbb{H}^n(X):=\mathbb{H}^0(X[n])\cong(\tau_{\ge n}\tau_{\le n}X)[n]$](./latex/latex2png-PerverseSheaves_40542036_-6.gif)
. 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
![$\mathbb{H}^{-1} Z=\tau_{\le0}(Z[-1])\cong \ker f$](./latex/latex2png-PerverseSheaves_31023171_-6.gif)
. 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
![$i^*j_*\mathcal{F}_U=i^!j_!\mathcal{F}_U[\pm1]$](./latex/latex2png-PerverseSheaves_205480802_-5.gif)
.
¡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
![$f_*\underline{k}_X[\dim X]\in P(Y)$](./latex/latex2png-PerverseSheaves_115940130_-5.gif)
.
![$H^i(f_*\underline{k}_X[\dim X])$](./latex/latex2png-PerverseSheaves_175851867_-5.gif)
has stalks

. So it is zero if

. So
![$$\dim\mathrm{supp}H^i(f_*\underline{k}_X[\dim X])\le \dim\coprod_{n>(i+\dim X)/2} Y^n\le-i$$](./latex/latex2png-PerverseSheaves_166702837_.gif)
as

. Namely
![$f_*\underline{k}_X[\dim X]\in {}^p\mathcal{D}^{\le0}$](./latex/latex2png-PerverseSheaves_55235735_-5.gif)
. One can similarly check that
![$\mathbb{D}f_*\underline{k}_X[\dim X]\in {}^p\mathcal{D}^{\le0}$](./latex/latex2png-PerverseSheaves_59307537_-5.gif)
.
¡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
![$$\mathcal{H}=\bigoplus_{x\in W}\mathbb{Z}[v^{\pm}] T_x,$$](./latex/latex2png-PerverseSheaves_87242300_.gif)
where
(Iwahori)
Let

,

. Then
![$\mathbb{Z}[B_q\backslash G_q/B_q]\cong \mathcal{H}_{\frac{1}{q}}$](./latex/latex2png-PerverseSheaves_63911001_-10.gif)
. ??
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

.
![$\Delta_w=(i_w)_!\underline{\mathbb{Q}}_{BwB/B}[\ell(w)]$](./latex/latex2png-PerverseSheaves_251279790_-10.gif)
, 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
![$\mathrm{ch}(\pi_s^*\pi_{s*})\mathcal{F}[1 ]=\mathrm{ch}(\mathcal{F})C_s$](./latex/latex2png-PerverseSheaves_142432386_-5.gif)
.
Now we will use the Push-Pull Lemma to prove the main theorem, namely
.