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 .