These are my live-TeXed notes for the course Math 268x: Pure Motives and Rigid Local Systems taught by Stefan Patrikis at Harvard, Spring 2014.
Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!
01/28/2014
Motivation
Let
be a field with algebraic closure
(so
). Consider smooth projective varieties
over
(either dropping the word smooth or projective will force us to enter the world of mixed, rather than pure, motives). There are several nice cohomology theory.
- the Betti realization
(a
-vector space), the singular cohomology of the topological space
.
- the de Rham realization
(a
-vector space with a Hodge filtration), the algebraic de Rham cohomology = the hypercohomology of the sheaf
of algebraic differential forms on
.
- the
-adic realization
(a
-vector space with
-action), the
-adic etale cohomology.
It is not even clear a priori that these
-vector space,
-vector space and
-vector have the same dimension. But miraculously there are comparison isomorphisms between them. For example,
(Comparison for B-dR)
There are isomorphisms
These isomorphisms are functorial and satisfy other nice properties (indeed an isomorphism of Weil cohomology, more on this later). This suggests that there is an underlying abelian category (of pure motives) that provides the comparison between different cohomology theory.
Slogan "sufficient geometric" pieces of cohomology have comparable meaning in all cohomology theory
We will spend a great amount of time on the foundation of all these different cohomology theory. But notice the comparison isomorphisms already suggest the various standard conjectures, for examples,
- Standard conjecture D: numerical equivalence = cohomological equivalence
- Standard conjecture C: Kunneth (the category of pure motives is graded and has a theory of weights).
- Standard conjecture B: Lefschetz (the primitive cohomology should be "sufficiently geometric")
Why should one care about the existence of such a category?
One motivation is that one gets powerful heuristic for transferring the intuition between different cohomology theory.
By the early 60's, one knew that if

is a smooth projective variety, then it follows from Hodge theory that

naturally carries a
pure Hodge structure of weight

, i.e. a

-vector space

with a bi-grading

such that

, where

is the complex conjugation with respect to

. On the other hand, Weil had conjectured that for a smooth projective variety

. The

-representation

is
pure of weight

, in the sense that the eigenvalues

of the geometric Frobenius

are algebraic numbers and for each embedding

of

into the complex numbers,

.
When
is smooth but not projective, people played with examples and found that
can be filtered (the weight filtration) such that the Frobenius eigenvalue is pure on each graded piece.
Let

be a smooth projective curve,

be a finite set of points and

. Then we have an exact sequence

Here

is pure of weight 1 and

(

, the

-adic cyclotomic character) is pure of weight 2 and

is also pure of weight 2. Therefore one obtains an increasing weight filtration on

:
The above mentioned
-adic intuition (generalized to higher dimension) lead Deligne to mixed Hodge theory. To give
a mixed Hodge structure for
not smooth projective, the key point is to find a spectral sequence
such that its
term is (conjecturally) pure of weight
.
In Hodge II, Deligne treated the case of smooth but no longer projective varieties
. The (
-adic analogue of the) spectral sequence is the Leary spectral sequence for
, where
a smooth compactification of
with
is a union of smooth divisors with normal crossings,
One can explicitly compute the sheaf
where
is smooth. Therefore
is pure of weight
.
Let us look at the differential
: notice both the target and the source are pure of weight
(all
are pure of weight
), nothing is weired. But on the
-page,
, where the source has weight
and the target has weight
respectively. The mismatching of the weight of the Frobenius eigenvalues implies that
for
. Therefore the Leray spectral sequence degenerates at
-page. One can compute that
The Betti analogue (of maps of pure Hodge structure) is provided by the reinterpretation that
and the differentials
's are simply Gysin maps ( = Poincare dual to pullbacks), which are also maps of pure Hodge structures.
The upshot is that the
-adic Leray spectral sequence gives the weight filtration (= the Leary filtration up to shift), and the graded piece
is pure of weight
. The Betti Leray sequence also gives a weight (defined to be) filtration on
such that we already know that
are naturally pure Hodge structures.
Another motivation for considering the category of pure motives is toward a motivic Galois formalism.
Let

be a field. The classical Galois theory establishes an equivalence between finite etale

-schemes with finite sets with

-actions. Linearizing a finite set

with

-action gives finite dimensional

-vector spaces

with the continuous

-action

. The linearization of finite etale

-schemes are the
Artin motives (motives built out of zero dimension motives). The equivalence between the two linearized categories is then given by

.
Generalizing to higher dimension: the category of finite etale
-schemes is extended to the category of pure homological motives. The Standard conjectures then predict that it is equivalent to the category of representations of a certain group
, which is a extension of classical Galois theory
These are still conjectural. But one can replace the category of pure homological motives by something closely related and obtain unconditional results. In this course we will talk about one application of Katz's theory of rigid local systems (these are topological gadgets but surprisingly produce motivic examples): to construct the exceptional
as a quotient of
(the recent work of Dettweiler-Reiter and Yun).
01/30/2014
Weil cohomology
We now formulate the notion of Weil cohomology, in the frame work of motives.
Let

be a field. Let

be smooth projective (not assumed to be connected) variety over

. Let

be the category of such varieties. Then

is a symmetric monoidal via the fiber product

with the obvious associative and commutative constraints and the unit

.
Let

be a field. Let

be the category of finite dimensional graded

-vector spaces in degrees

with the usual tensor operation

It is endowed with a graded commutative constraint via
A
Weil cohomology over

(a field of characteristic 0) on

is a tensor functor

, namely,

comes with a functorial (Kunneth) isomorphisms

respecting the symmetric monoidal structure. Notice the monoidal structure induces a
cup product 
making

a graded commutative

-algebra. We require it to satisfy the following axioms.
- (normalization)
. In particular,
is invertible in
. We define the Tate twists
(this is well motivated by
-adic cohomology).
- (trace axiom) For any
of (equi-)dimension
, there is a trace map
satisfying
- Under
, one has
.
and the cup product induces a perfect duality (Poincare duality) 
- (cycle class maps) Let
be the
-vector spaces with a basis consisting of integral closed schemes
of codimension
. Then there are cycle class maps
satisfying
factors through the Chow group
(modulo the rational equivalence).
is contravariant in
, i.e., for a morphism
and a cycle
of codimension
, we have
whenever this makes sense. This will always make sense after passing to the Chow group. In general, one cannot always define
on
. But if
is flat , then one can: in fact, by flatness
has all its components of codimension
in
(but
is not necessarily integral). Let
be the (reduced structure) of the irreducible components. One then associates a cycle
where
, the length of the local ring at
. We also require it to be compatible with pushforward (defined in Definition 10) that 
- For
,
.
Notice
is not necessarily a combination of integral closed subscheme (e.g.,
is nonreduced), the cycle
should be understood as the reduced structure with multiplicity.
- (pinning down the trace) the composite
sends
to
, where
are closed points.
(Trace in Betti cohomology)
Let

be smooth projective of dimension

. Define

to be the composite
![$$\scriptstyle H^{2d}_B(X)(d)\xrightarrow{\frac{1}{(2\pi i)^d}} H^{2d}_B(X)\xrightarrow[\alpha]{\sim} H^{2d}(X, \underline{\mathbb{Q}})\rightarrow H^{2d}(X,\underline{\mathbb{Q}}) \otimes \mathbb{R} \xrightarrow[\beta]{\sim} H^{2d}(X,\mathcal{A}_X^\cdot)\xrightarrow{\int_X}\mathbb{R},$$](./latex/latex2png-PureMotives_229627239_.gif)
where

is the smooth de Rham complex with

-coefficients. Notice

is an isomorphism because the sheafy singular cochain complex is a flasque resolution of

and

is an isomorphism because

is a fine resolution of

. The choice

will chancel out the choice of the orientation we made on complex manifold when we do integration and one can check that

lands in

.
Algebraic de Rham cohomology
Suppose
is a field of characteristic 0 and
smooth (not necessarily projective).
We define the
algebraic de Rham cohomology 
to be the hypercohomology of the sheaf

of algebraic differential forms on

.

is a Weil cohomology.
We now selectively check a few of the axioms.

is a tensor functor.
Given a morphism

. Let

be an injective resolution of

and

be an injective resolution of

. Then

is quasi-isomorphic to

. The map

induces the pull-back on
The Kunneth isomorphism
is explicitly given by
, where
and
are the natural projections.
¡õ
When

is affine, we have

(this follows from the vanishing of

for

affine and

quasi-coherent; in particular, ), which makes the computation feasible. For example, the de Rham complex for

is simply
![$k[T]\xrightarrow{d}k[T]dT$](./latex/latex2png-PureMotives_29759566_-5.gif)
. So taking cohomology gives

and

. This also gives an example in characteristic

that

is infinite dimensional because

, so one don't really want to work with the algebraic de Rham cohomology in characteristic

!
02/04/2014
In general, one covers
by open affines
. For any quasi-coherent sheaf
on
, one then obtains the Cech complex
, a resolution of
by acyclic sheaves, defined by
Now we have a double complex
whose columns are acyclic resolutions of
. The general formalism implies that
the cohomology of the global sections of the total complex. Recall the total complex is defined by
where
.
(normalization)
Let

and

be a covering of

. The Cech double complex looks like
![$$\xymatrix{ \mathcal{O}(U_0\cap U_1)\ar[r] & \Omega^1(U_0\cap U_1)\\ \mathcal{O}(U_0)\oplus\mathcal{O}(U_1) \ar[r] \ar[u] & \Omega^1(U_0)\oplus\Omega^1(U_1) \ar[u]}$$](./latex/latex2png-PureMotives_28170513_.gif)
The total complex is thus

where the two differentials are given by

and

One can easily compute

,

and

is 1-dimensional generated by

. Indeed one sees the computation really shows

This is an instance of the Hodge to de Rham spectral sequence.
If

and

is projective, then the Hodge to de Rham spectral sequence degenerates at the

-page.
To define the trace for the algebraic de Rham cohomology, we proceed in two steps. We first show that
is abstractly the right thing, i.e.,
and is
if
is geometrically connected. Then we pin down that actual map
after defining the de Rham cycle class map.
The first step uses the Serre duality. By the Hodge to de Rham spectral sequence, we have a map
By Serre duality (
is the dualizing sheaf), one has the trace map
So we want to say that the map
is an isomorphism. By the Hodge de Rham spectral sequence, it is enough to show that
(or
because
is a free
-module of rank 1). This can be checked bare-handed by reduction to
: choose a finite flat map
to get the trace map
(so
). It follows that
is injective. It suffices to prove that
which boils down to the direct computation that
using the Hodge to de Rham spectral sequence.
To pin down
, we need to choose carefully a generator
of
and set
i.e.,
.
De Rham cycle class maps (via Chern characters)
We seek a cycle class map such that
as follows.
- Define the Chern class
of line bundles.
- Define the Chern class
of vector bundles.
- Define the Chern character
of vector bundles on
.
- One knows that
factors through
, Grothendieck group of vector bundles on
. Using the fact that
is smooth, the latter can be identified with
, the Grothendieck group of coherent sheaves on
.
- For
a codimension
cycle,
makes sense and we define the cycle class map 
Now we describe each step in details.
Step a
We want a group homomorphism
. Identify
. The map
induces a map
and hence induces the desired map
.
Step b
Let
be a vector bundle of rank
on
. Denote the projective bundle
. Notice
on
has a tautological line subbundle
. Let
. The fact (the Leray-Hirsch theorem, a special case of the Leray spectral sequence) is that
is a free module over
with basis
. We then define
by
Notice this agrees with the previous definition of
of line bundles and is functorial in
.
02/06/2014
Define the total Chern class
The key is the following multiplicative property.
For any sort exact sequence of vector bundles

we have

.
To show this, one first show that if

is a direct sum of line bundles, then

Then reduce the general to the first case by showing that there exists a map

such that

splits as direct sum of line bundles and

is injective on

(the
splitting principle).
For the first case, since the statement is invariant under twist, one can assume each
is very ample of the form
and reduce to the case to the case of
being a product of projective spaces. Notice that each
gives a section
and by definition
. Write
. Pullback the defining relation for
along each
, we obtain the relation
So the polynomial
in
has roots
. But
has the advantage of being like a polynomial ring,
Assume
, then the defining relation must be
, which shows that
is the
-th symmetric polynomial of
as desired.
To reduce the general case to the first case, arrange
so that
has a full flag of subbundles by iterating the projective bundle construction, then split the extension by further pullback: if one has a surjection
of vector bundles, then the sections
form an affine bundle over
; pulling back along this affine bundle splits
and induces an isomorphism on cohomology.
¡õ
Step c
Using the multiplicative property, we can define formally the Chern roots
of
so that
. Here the Chern roots don't not make sense but their the symmetric polynomials do make sense in cohomology. Define the Chern character
This makes sense in cohomology. Now we have the additivity
in exact sequences. Moreover
. Therefore we obtain a ring homomorphism 
When
is smooth, one can form finite locally free resolutions of any coherent sheaves on
, and taking the alternating sum of the terms in the resolutions induces the inverse of natural map
. Thus
(see Hartshorne, Ex III.6.8).
Step d
For
a codimension
cycle,
makes sense and we define the cycle class map
In particular, our choice of the basis
for
is given by for any closed point
of
,
This is the choice we made to normalize the trace map. We need to check that
is independent on the choice of
(this follows from connecting two points by a curve in
and the invariance of
in a flat family). We also need to check that
. This reduce to the case of projective spaces. Let
be a closed point. One can put
in a chain
Using the short exact sequences of the form (given a choice of a section of
),
for each
, it follows that in
, we have
Applying the Chern character we obtain that for
,
which is nonzero.
Formalism of cohomological correspondences
Let
be a Weil cohomology.
Given a morphism

, we define the
Gysin map 
to be the transpose of

under the Poincare duality. At the level of cycles,

is basically

when

and zero otherwise (this matches the degree shift in

).
(Projection formula)
Let

,

, then

.
The property

characterizes

.
¡õ
A
cohomological correspondence from

to

is an element

interpreted (using the Poincare duality and the Kunneth formula) as a linear map

. Explicitly, if

, then

(extended to be zero away from top degree). Let

,

be the natural projections. Then another way of writing

is

Namely, pullback

, intersect with

, then pushforward to

. One can check that

by the projection formula.
Define

,

.
The
transpose 
of

is defined to be the image of

in

under

. One can check that

.
(Composition of correspondences)
For

,

, we define

.

.
Notice that

. The claim then follows from the
associativity of composition of correspondences. For details, see Fulton, Intersection theory, Chapter 16.
¡õ
Let

be the graph of the morphism

. Then

,

,

.
Our next goal is to deduce the Weil conjecture (except the Riemann hypothesis) from a Weil cohomology (hence the name). We will later see that the Riemann hypothesis follows from the standard conjectures.
02/11/2014
Formal consequences of a Weil cohomology
Let
be a Weil cohomology.
(Lefschetz fixed point)
Suppose

is algebraically closed. Let

be connected. If

,

are of degree

and

respectively (namely,

and induces

; similar for

). Then
We compute by each Kunneth component so let

,

. Let

be a basis of

and let

be a dual basis of

such that

. So we can write

Here

So the left hand side is equal to

Switching

and

introduces another sign

which cancels out the sign

since

. So the left hand side is equal to

To compute the trace on the right hand side, we notice that

Since we care only about the

-term when taking the trace, this matches the left hand side.
¡õ
Let
be a cohomological correspondence so that
on
. Write
where
is the cohomological correspondence
. So
.
Let

(so

is of degree zero). Then
Taking
and using
, we obtain the following refinement.
Now let
and
be the (absolute) Frobenius morphism. Then
is the fixed point of
for any
.
(Grothendieck and others)
There exists a Weil cohomology on

,

.
To interpret the left hand side as the fixed points of
, we need the following lemma.

and

intersect properly: every irreducible component of

is of codimension

(i.e., the codimensions add). So

can be computed as a sum of local terms, one for each point in

. Moreover, the local terms are multiplicity-free (by computing the tangent space intersection

at an intersection point

).
Therefore we conclude that 
The Weil conjecture (expect the Riemann hypothesis) the follows.
The zeta function

can be computed as
The previous corollary of the Lefschetz fixed point theorem and the easy linear algebra identity

proves the claim.
¡õ
Combining this cohomological expression of
with the Poincare duality, we also obtain the functional equation of
(part of the Weil conjecture).

Here

is the Euler characteristic of

.
Intersecting cycles
Let

be a smooth quasi-projective variety. For any
proper (this is not serious since we will be working in

), we define pushforward cycles by
![$f_*(Z)=[k(Z):k(f(Z))]f(Z)$](./latex/latex2png-PureMotives_11322962_-5.gif)
when

and 0 otherwise.
On the other hand, we defined pullback of cycles along a flat morphism
(Definition 3 c)).
We would like to make sense of pullback for more general classes of morphisms. Moreover, such pullback should be compatible with the pullback on cohomology under the cycle class maps. This can be done if there is a cup product (intersection pairing) on the group of cycles,
by intersecting with the graph of
. This is not naively true since the two cycles may not intersect properly (the codimension is wrong). So first we restrict to properly intersecting cycles
whose intersection
has all components of the right codimension. Then
should be a sum of irreducible components of
with multiplicities
here
is the local ring of
at the an irreducible component of the intersection
. This formula of intersection multiplicities (due to Serre) defines an intersection product for properly intersecting cycles.
To deal the general case, the classical approach is to jiggle
to make the intersection properly meanwhile staying in the same rational equivalence class (moving lemma).
We say two cycles

of dimension

are rationally equivalence if

is generated by terms of the following form. Let

be a

-dimensional closed subvariety and take its normalization

; these generators are the proper pushfowards

for

.
An alternative approach is to consider
a
dimension closed subvariety. Then the rationally equivalent to zero cycles are generated by
, here
is the fiber of
in
.
These two definitions are equivalent. One can check that being rational equivalent is a equivalence relation. We denote it by
.
The
Chow group (with

-coefficient)

.
Chow's Moving Lemma then gives a well defined intersection pairing on the Chow groups
This makes
a graded and commutative unital ring. The proper pushforward
descends to the level of Chow groups.
We define the pullback

on Chow groups for

proper by

.
02/13/2014
Adequate equivalences on algebraic cycles
We showed last time that

on

is an adequate relation.
For any Weil cohomology

, the
cohomological equivalence 
is an adequate relation. Here

if

in

. Notice that a priori these cohomological equivalences may not be independent of the choice of

. If two such Weil cohomology theories are related by comparison, e.g.,

and

, then the corresponding cohomological equivalences are the same.
We say

is
numerically equivalent 0 if for all

,

, here the degree map

,
![$\sum n_iP_i\mapsto\sum n_i[k(P_i):k]$](./latex/latex2png-PureMotives_102579138_-5.gif)
(one can think of it as

, for the structure map

). Then

is an adequate relation.
is the finest adequate equivalence relation.
is the coarsest adequate equivalence relation.
- Let
be an adequate relation. We want to show that if
, then
. By definition,
is linear combination
. Let
and
be the projections. Then
Suppose we knew that
. Then by the definition of adequate relation
, we know
. So we reduced to show that
on
. Let
(assume
for simplicity). Since
is adequate, we can find
intersecting properly with
, i.e.,
with
. We can certainly write down a map
such that
and
. Explicitly,
Therefore we have a chain of equivalences
as desired.
- The second part is basically a tautology.
¡õ
Let

be an adequate equivalence relation on

. Let

be field of characteristic 0 (e.g.,

). We define

, the ring of cycles on

modulo

.
Let

be the category with objects

(usually write it as

thought as a cohomological object), and

(Think: graphs of homomorphisms

.) This is an

-linear category, with

There is a functor
We want to enlarge
to include images of projectors. There is a universal way of doing this by taking the pseudo abelian envelope. We also want duals to exist in our theory (this amounts adding Tate twists). Combining these two steps into one,
We define the category

(the coefficient field

is implicit) of
pure motives over
modulo 
. Its object is of the form

, here

is an idempotent in

and

is an integer (Think:

). The morphisms are given by

Here the existence of Tate twists allows one to shift dimensions (e.g, a map

.
is pseudo abelian ( = preadditive and every idempotent has a kernel).
is
-linear. The addition is given by (if
)
Here we think of
as the summand
of
and identify 
- (next time) There is a
-structure 
Grothendieck conjectured (Standard Conjecture D) that for
,
for any Weil cohomology
. He also conjectured that
is abelian. Hence under Conjecture D,
is abelian. Conjecture D is still widely open, but in the early 90s, Jannsen proved the following startling theorem.
That means that the numerical equivalence is arguably the "unique" right choice for the theory of motives.
02/18/2014
Tannakian theory
Let
be an additive tensor ( = symmetric monoidal) category. One can check for the unit object
, then endomorphisms
is a commutative ring and
becomes an
-linear category.
We say a category

is
rigid if for any

there exists

("dual") and morphisms

such that the composite map

is

and the composite map

is

.
Let

be a field. A
neutral Tannakian category 
is a rigid abelian tensor category with

and for which there exists a fiber functor

. By a
fiber functor, we mean a faithful, exact,

-linear tensor functor. It is neutralized by a choice of such a fiber functor. (Think: the category of locally constant sheaves of finite dimensional

-vector spaces on a topological space

; a fiber functor is given by taking the fiber over

).
The main theorem of Tannakian theory is the following.
Let

be a neutral Tannakian category over

and let

be a fiber functor. Then the functor on

-algebras

is represented by an affine group scheme over

and

is an equivalence of categories.
02/25/2014
The Kunneth Standard Conjecture (Conjecture C)
Suppose

. Assume

(Conjecture D) and that all Kunneth projectors

are all algebraic cycles (Kunneth). Then

(with

-coefficients) is an a neutral Tannakian category over

.
By Jannsen's theorem,

is abelian. We saw last time that

with its given naive commutative constraint could not be Tannakian. So we will keep the same tensor structure but modify the commutativity constraint using Kunneth. Kunneth tells us that

is

-graded via the projectors

, i.e., for any

, we get a weight decomposition

Now for any

, we define the modified commutativity constraint

given by

Now

is a fiber functor.
¡õ
- For any
, and
an abelian variety,
is true.
- For
a finite field, then
is true for any
(with respect to any Weil cohomology satisfying weak Lefschetz). This is a theorem of Katz-Messing. Deligne's purity theorem on
allows one to distinguish different degrees. Katz-Messing shows that for any Weil cohomology with weak Lefschetz, the characteristic polynomial
of the Frobenius
on
agrees with that on the
-adic cohomology. Choose a polynomial
such that
(for
) and
, then
is algebraic (as the combinations of the graphs of
) and is the projection onto
.
The Lefschetz Standard Conjecture (Conjecture B)
Let

be a Weil cohomology. We say

satisfies the hard Lefschetz theorem if for any

, any ample line bundle

and any

,

is an isomorphism. Here

.
When

and

, this is part of Hodge theory. For any

and

, this is proved by Deligne in Weil II.
The hard Lefschetz gives the primitive decomposition of
.
For all

, define

(this depends on the choice of

). Then
One should think of
as a nilpotent operator on
, then the Jacobson-Morozov theorem implies that this action can be extended to a representation of
. The primitive parts are exactly the lowest weight spaces for this
-action.
Let
. Then
is semisimple and
sends
to
. So applying Jacobson-Morozov gives a unique
-triple
(the name
comes from Hodge theory). Moreover, it follows that
.
02/27/2014
A more convenient operator, the
Hodge star 
, can be extracted as follows. The

-action on

gives rise to a representation of

on

. Suppose

is the weight

eigenspace for

. Then

is in the

-eigenspace. But

is not quite an involution. So we renormalize and define

on

and then

.
Another variant is the
Lefschetz involution 
for

. Then

as well. It differs from

from certain rational coefficients on each primitive component.
Now we have the following cohomological correspondences:
,
,
,
,
(
is the inverse to
on the image of
),
- Kunneth projectors
,
- Primitive projectors
:
- For
,
for
and 0 on
;
- For
,
for
(so it satisfies
).
The following lemma is immediate.

,

,

,

,

,

are all given by universal (noncommutative) polynomials in

and

.
One can show that
![$p^d,\ldots, p^{2d}\in \mathbb{Q}[L,\Lambda]$](./latex/latex2png-PureMotives_244525707_-5.gif)
.
¡õ
Now we can state various versions of the Lefschetz Standard conjecture.
(Weak form
)
For

,

is an isomorphism (i.e., it is surjective).
(Strong form
)
The operator

is algebraic. Namely, it equals to the cohomology class a cycle in

.
Because
, we know that

.
Under

and

, if

is algebraic and induces an isomorphism

. Then

is also algebraic (see Remark
14).
Notice

gives a map

. Under

and

, this map is algebraic and an isomorphism. Hence

is an algebraic and an isomorphism. Therefore

is algebraic by Remark
14, so

is also algebraic.
¡õ

is independent of the choice of the ample line bundle

giving rise to

.
Suppose

is given by another ample line bundle

. Then the hard Lefschetz tells us that

is an algebraic isomorphism (notice the correspondence

is equal to

, hence

is algebraic when

is algebraic). Hence its inverse is also algebraic by the previous corollary. Now use e) of the previous proposition.
¡õ
The Hodge Standard Conjecture (Conjecture I)
The standard conjectures B and C both follow from the Hodge conjecture. The only standard conjecture does not follow from Hodge conjecture is the Hodge Standard conjecture. It concerns a basic positivity property of motives.
Take
. For any
,
carries a pure
-Hodge structure of weight
. More fundamental in algebraic geometry is the polarizable
-Hodge structure.
For

and

an ample line bundle, we have

The class

can be thought of as the Kahler form in

(valid for general Kahler manifolds). Define

Extending

-linearly we define the sesquilinear pairing
We would like to study the positivity properties of
by reducing to particular pieces of the bigrading and the primitive decomposition.
(Hodge index theorem)
On

,

is definite of sign

.
On a curve

,

has sign

on

and

on

. Suppose

. Then
On a surface

,

on

has sign

, and sign

on

. So

is negative definite on

, positive definite on

and positive definite on

. For example, if

is a K3 surface, then

has signature

on

and has signature

on

.
This theorem is the source of polarization in Hodge theory.
A weight

-Hodge structure

(

is a

-vector space,

) is
polarizable if there exists a morphism of Hodge structures

such that

is positive definite.
So the Hodge index theorem has the following corollary.
For any

,

is a polarizable

-Hodge structure. A polarization is given by

, where
03/04/2014
Let

. We need to show that

satisfies

is positive on

. Let

Then

Now using the Hodge index theorem we see the sign cancels out and takes value in

.
¡õ
Now we would like a (weak) version of this that makes sense for any field
and any Weil cohomology
satisfying hard Lefschetz (so the primitive cohomology
still makes sense). Inside
there is
-vector subspace
.
(Hodge Standard Conjecture
)
For any

, the pairing on

given by

is positive definite.
By Corollary 10(take
),
For

and

,

holds unconditionally.
We now explain that for
, the Hodge Standard conjecture implies the Riemann hypothesis. A more convenient reformulation of
is that the pairing
is positive definite. It follows that there is a positive involution on
(acting on
) given by
Explicitly,
(which is algebraic under Lefschetz).
So we want that the eigenvalues of the Frobenius
on
are pure of weight
. We renormalize the Frobenius (acting on
) as
Under Lefschetz,
. We want all eigenvalues of
has absolute value 1 for all complex embeddings. This can be obtained by realizing
as a unitary operator on the inner product space (
). We notice that
commutes with
and
, so
We claim that
, so that
is
-invariant. This follows from the following more general lemma. One can check that
(
is the chosen ample line bundle), so the following lemma applies to
.
b), c) implies that

. Therefore

is invertible. In fact, for

nonzero, find

such that

, then

has nonzero trace, so

. Now

So

.
¡õ
It follows that
is unitary with respect to the inner product
(the positivity follows from
and the fact that
). In particular, the eigenvalues
acting on
have all absolute values 1. Hence by Cayley-Hamilton, the roots of characteristic polynomials of
on
have all absolute values 1, as desired.
Absolute Hodge cycles
Our next goal is to construct a modified category
of pure motives such that
- Under the standard conjectures,
.
has all categorical properties we want: (say
) it is
-linear, semisimple, neutral Tannakian (this gives unconditional motivic Galois formalism).
lets you prove some unconditional results and formulate interesting but hopefully more tractable than the standard conjecture problems.
The basic strategy is to redefine correspondences using one of these larger classes of cycles:
algebraic cycles
motivated cycles (Andre)
absolute Hodge cycles (Deligne)
Hodge cycles
An
absolute Hodge cycle on

(suppose

has characteristic 0 and finite transcendence degree) is a class

in

where

, such that for all

, the pullback class

comes from a Hodge cycle in

(a

-vector space) via the comparison isomorphisms.
(Deligne)
Any Hodge cycle on an abelian variety (

) is absolutely Hodge.
One should think of this as a weakening of the Hodge conjecture for abelian varieties.
We will define Andre's notion of motivated cycles next time. Along this line,
(Andre)
Any Hodge cycle on an abelian variety (

) is motivated.
One classical application of absolute Hodge cycles is the algebraicity of (products of) special values of the

-function like

(with refinements giving the Galois action). The origin of this comes the periods (i.e. coefficients of the matrices in the B-dR comparison theorem) of the Fermat hypersurface

For an algebraic cycle

(defined over

) and a differential form

such that
![$[\omega]\in H^*_\mathrm{dR}(X/k)$](./latex/latex2png-PureMotives_165369171_-5.gif)
, then one obtains a period

The same principle applies for

an
absolute Hodge cycle. A good supply of absolute Hodge cycles for Fermat hypersurfaces are the Hodge cycles by Deligne's theorem for abelian varieties (the motive of Fermat hypersurfaces lie in the Tannakian subcategory generated by Artin motives and CM abelian varieties).
03/06/2014
More generally, let
be a number field and
a smooth projective variety. Let
be the field generated by the coefficients of the period matrix. The relations between periods are predicted by the existence of algebraic cycles. The transcendence degree of
is equal to the dimension of the motivic Galois group (when one makes sense of it). For the motive
(defined by absolute Hodge cycles), we have
. Deligne's theorem implies that the later is equal
, the dimension of the Mumford-Tate group (the Hodge theoretic analogy of the motivic Galois group).
Let

be a

-Hodge structure. The
Mumford-Tate group 
is the

-Zariski closure of the image of

(i.e., the smallest

-subgroup of

whose

-points containing

.

;

.

;

.
Notice a priori, one only knows the inequality
(since absolute Hodge cycles
Hodge cycles).
Here is another application due to Andre. Suppose

is finitely generated. Let

,

are K3 surfaces over

with polarizations (the important fact is that

for K3 surfaces). Then any isomorphism of

-modules

arises from a

-linear combination of
motivated cycles. Also the Mumford-Tate conjecture is true for

: namely,

is equal to the connected component of the Zariski closure of the image of

on

. This is not known even for abelian varieties: there are a lot of possibilities of Mumford-Tate groups for abelian varieties, but for K3 surfaces they are quite restricted. Let

be the orthogonal complement of Hodge cycles (the
transcendence lattice which is 21 dimensional generically). Then

is a field because

, and is either totally real or CM due to the polarization. A theorem of Zarhin shows that in the totally real case the Mumford-Tate is a special orthogonal group over

and in the CM case a unitary group over

, with the pairing coming from the polarization.
Motivated cycles
Let
be a Weil cohomology with hard Lefschetz. Fix a subfield
(e.g.
).
(Motivated cycles)

is defined to be the subset of elements of

of the form

for any

and any

algebraic cycles. Here

is the Lefschetz involution associated to a product polarization
![$\eta_X \otimes [Y] + [X] \otimes \eta_Y $](./latex/latex2png-PureMotives_157620616_-5.gif)
on

. The idea is that we don't know Lefschetz and so we manually to add all classes produced by the Lefschetz operators to algebraic cycles to get motivated cycles.
The basic calculation (with the above remark) shows the following.
is an
-subalgebra of
(with respect to the cup product).
.
.
As for algebraic cycles, we define the motivated correspondences similarly.
Define

with the similar composition law (the target is correct by the previous lemma). Then

is a graded

-algebra. We also have a formalism of

and projection formulas for

.
(analogue of Jannsen's theorem)
For any

,

is a finite dimensional
semisimple 
-algebra, hence

is semisimple abelian.
We define an analogue of numerical equivalence:

is called to be motivated numerically equivalent to 0 if for any

,

. Then Jannsen's argument shows that

is semisimple. But since

and

holds for motivated cycles by construction, the motivated equivalence is the same as the motivated numerical equivalence (Remark
22). Therefore

is semisimple.
¡õ
03/11/2014
(Properties of
)
is pro-algebraic, even pro-reductive over
.
splits over the maximal CM extension of
(i.e., for any
and
,
is isomorphic to
for some
, where
is the maximal CM subfield).
The Motivated variational Hodge conjecture
Source of motivated cycles: the motivated analogue of variational Hodge conjecture.
(Variational Hodge conjecture)
Over

, let

be a smooth projective morphism, let

. If

is algebraic for some

, then for any

,

is also algebraic.
(Andre)
The variational Hodge conjecture holds with "motivated" in place of "algebraic".
Let us review the theorem of the fixed part and the necessary background in mixed Hodge theory.
Suppose

is smooth projective and

is smooth. Let

be a smooth compactification. Then

is surjective. In other words, the image

is the fixed part under the monodromy, i.e.,

.
The above maps are given by

Here
is the edge map in the Leary spectral sequence associated to
. By a theorem of Deligne, when
is smooth projective, the Leary spectral sequence degenerates at
. So
is surjective.
and
have the same image. Since
is injective, it follows that
and
have the same image. Hence
is surjective.
More generally, if
is smooth projective (applied to
),
is smooth, Then the image of the composite map
is the same as the image of the latter map. The reason is that each of these cohomology groups has a weight filtration
such that
is a pure Hodge structure of weight
. Since
,
are smooth, their weight filtration looks like
Since
is smooth but not projective, its weight filtration looks like
The general important fact is that the morphisms of mixed Hodge structure are strict for the weight filtration (one consequence: mixed Hodge structures form an abelian category), i.e., for
of mixed Hodge structure, then for any
,
Now the strictness implies that it suffices to check the images on each
are the same. The results then follows from
. To see this, it essentially follows from the definition of the weight filtration as the shift of the Leray filtration associated to
:
and by definition
is the whole thing.
¡õ
Let

be a motivated cycle such that a finite index subgroup of

acts trivially on

, then all parallel transport of

are still motivated.
Apply the previous theorem after a finite base change.
¡õ
Let

be an abelian variety. Then the Hodge cycles on

are known to be motivated, due to Deligne-Andre. The idea of the proof is to put

in a family with the same generic Mumford-Tate group, prove for Hodge cycles special abelian variety in the family and then use the variational Hodge conjecture. More precisely, any Hodge cycle

on

has the form

, where we can take

to be the product of an abelian variety and abelian schemes over smooth projective curves. So the Hodge conjecture for abelian varieties (not known) reduces to the Lefschetz standard conjectures for
abelian schemes over smooth projective curves.
For any abelian variety

,

.
This follows from Hodge cycles on abelian varieties are motivated and that the product of abelian varieties are still abelian varieties:

.
¡õ
Mumford-Tate groups
- Let
be the stabilizer of
. Then
is a sub Hodge structure if and only if
stabilizes
if and only if
factors through
if and only if
.
- Apply the first part to the subspace
.
¡õ
The natural functor from

to the category of

-Hodge structures is fully faithful and realize

as the Tannakian group of

(as a subcategory

-Hodge structure).
The full subcategory of polarized

-Hodge structures

is semisimple.
When

is polarizable,

is a (connected) reductive group.
The connectedness follows from the definition. To show that

is reductive, we only need to exhibit a faithful and completely reducible representation of

. The standard representation

works: the subrepresentations exactly corresponds to the sub Hodge structures of

, whose complete reducibility is ensured by the previous lemma.
¡õ
When

is polarizable,

is exactly the subgroup that fixes all Hodge tensors.
03/13/2014
This follows from the following general results. Let

be a reductive group and

be a subgroup of

. Define

If

is reductive, then

(

a priori). The claim follows from taking

and

. For any

(reductive or not), by the theorem of Chevalley, there exists a representation

of

and a line

such that

is the stabilizer of

. If

is further reductive, there exists a

-complement

. Then

and

consists of the elements fixing any generator of this line. So

.
¡õ
Let

, then

giving rises to

is exactly the subgroup of

fixing all motivated cycles in all tensor constructions.
Because motivated cycles
Hodge cycles,
Let

, then

.
Does
even arises as
for some polarized
-Hodge structure
? This is at least necessary for it to be a motivic Galois group.
A semisimple adjoint group

is a Mumford-Tate group of a polarizable

-Hodge structure if and only if

contains a compact maximal torus.
Let explain the case when
is simple with compact maximal torus over
. Write
. Let
be a compact maximal torus, fixed by some Cartan involution
of
. The Cartan involution is essential for the polarization. Namely,
is an involution on
satisfying the following positive condition:
is positive definite. Decompose
into the
and
eigenspaces for
. Here
matches up with the Lie algebra of the maximal compact subgroup
. Now any
yields a polarizable
-Hodge structure on
if and only if
is a Cartan involution on
.
Let us write down
. Choose a cocharacter
such that
for any compact roots
and
for any noncompact roots
. Notice such cocharacters is in bijections with
. Extend
(trivial on
) to obtain
Then
acts on the root space
by
, which is 1 when
is compact and
when
is noncompact. Now use
is negative definite on
and positive definite on
. One knows that
gives a polarization on
. Using this framework, it is easy to check
can't arise as
.
Consider the split form of

.

. The two compact roots are

,

.
After the break we will construct
as a motivic Galois group via the theory of rigid local systems. This is originally due to Dettweiler-Reiter using Katz's theory. Zhiwei Yun gives an alternative proof (also for
and
). We will focus on the former, since the latter needs more machinery from geometric Langlands.
Applications of the motivated variational Hodge conjecture
The Kuga-Satake construction is "motivated", i.e., for

a projective K3 surface, the attached abelian variety

such that

which is a priori a morphism of

-Hodge structure, is indeed a motivated cycle (i.e., a morphism in

). This implies the Mumford-Tate conjecture for K3 surfaces, etc..
Now let us give the motivated analogue of a refinement of this assertion. So we need a notion of a family of motivated motives.
- Let the exceptional locus
does not contain the image of a finite index subgroup of
. Then
is contained in a countable union of closed analytic subvarieties of
.
- (Refinement) There exists a countable collection
of algebraic subvarieties
such that
is contained in the union of
. (In Hodge theory, this continues to hold for arbitrary
-polarized variational Hodge structure. This "algebraicity of the Hodge loci" is a strong evidence for the Hodge conjecture.)
- There exists a local system
of algebraic subgroups of
such that
for any
.
for all
.
contains the image of a finite index subgroup of
(notice the latter is a purely topological input!)
03/25/2014
Rigid local systems
Let
be a smooth projective connected curve. Let
be a finite set of points. Let
. For the time being, we work with the associated complex analytic spaces (so
implicitly).
A
local system 
of

-vector spaces on

is a locally constant sheaf of

-vector spaces.
The case

is most interesting for our purpose. Here

is a free group on

generators.
Given a local system

on

, when does
come from geometry?
By coming from geometry, we mean there exists a smooth projective family
such that
for some
(notice
itself is a local system).
One necessary condition for
to come from geometry is that the local monodromy at each puncture should be quasi-unipotent (some power of it is unipotent, equivalently, all its eigenvalues are roots of unity). This follows from the local monodromy theorem:
Any polarizable

-variational Hodge structure over a punctured disc

has quasi-unipotent monodromy.
The sufficient condition to come from geometry is still a total mystery. Simpson's guiding philosophy is that rigid local systems shall always come from geometry. Katz's book proves this is the case for irreducible rigid local systems on
.
There are several notions which you may want to call rigid local systems.
Let

be a

-local system on

.

is
physically rigid if for local system

such that for any

,

, then

. In terms of the monodromy representation: if the generators are conjugate (possibly by different matrices), then they are globally conjugate.
A slight variant:

is
physically semi-rigid if there exists finitely many local systems

such that if

is locally isomorphic to

for all

(as in the previous definition), then

for some

.
These two notions are very intuitive but extremely hard to check. The following definition provides a numerical condition and is easier to check.

is
cohomologically rigid if

. Here

. Notice

is still a local system on

(but

is no longer a local system on

).
The following lemma gives a very useful numerical criterion for cohomologically rigidity.
Let

be an irreducible local system of rank

on

. Then

is cohomologically rigid if and only if

, if and only if
03/27/2014
We denote the Jordan block of length

with the eigenvalue

by

. Take

,

and

. They give a local system on

. They have Jordan forms

,

and

, all are quasi-unipotent. It actually comes from geometry (classically known) as the local monodromies of the Legendre family

Namely, it comes from the local system

(e..g, one can see these matrices by Picard-Lefschetz). Moreover, it is cohomologically rigid by the previous lemma:

and each

.
Hypergeometric local systems are irreducible.
If not, let

be a subrepresentation and

be the corresponding quotient. Since

is a pseudo-reflection, we know that it must acts trivially on one of

, hence

on one of them, which contradicts the assumption

.
¡õ
Given
, one can write down the explicit matrix description for the local monodromies.
- It suffices to show that
is pseudo-reflection:
indeed has rank 1.
- Given such an
, Set
,
. Let
. Then
has dimension
since
is a pseudo-reflection). Hence
has dimension at least one; let
be a nonzero vector of this space. Thus
. Therefore
,
, ...,
. We claim that the span
is the whole space. In fact, by Caylay-Hamilton
stabilize on this span, so it must be the whole space by the irreducibility. In this basis,
,
,
have the desired form.
¡õ
- What are (the Zariski closure) of the monodromy group
of hypergeometric local systems? For example, does
appear?
- Are those with
roots of unity always geometric?
- We saw that hypergeometric local systems are both physically rigid and cohomologically rigid. What is the relationship between physical and cohomological rigidity in general?
- Not
. Beukers-Heckman computed all possibilities:
,
,
and some specific finite groups.
- Yes, by Katz's theory.
Now let us come to the third question in more detail.
Let

be an irreducible local system on

. Suppose

is cohomologically rigid, then

is physically rigid.
Let

be a local system with the same local monodromy as

. Since

and

has the same local monodromies, the Euler characteristic formula implies that

So

and by Poincare duality,

So at least one of the local systems

,

has a global section. Since

is irreducible and

, this global section gives an isomorphism

.
¡õ
For the other direction, we will use a transcendental argument. This direction is not known in the
-adic setting (knowing local monodromy matrices is not enough in the
-adic setting: one needs to know continuity).
Suppose

. Let

be an irreducible local system on

. Suppose

is physically rigid, then

is cohomologically rigid.
We know a prior that

. We need to show it is

. Let

be the local generators around the punctures. Suppose

is given by matrices

and

is given by matrices

. Since

is physically rigid, if there exists

such that

, then there exists

such that

. We want to show that

Consider the map

The fiber

corresponds to the local systems with the same local monodromies as

. The group

acts on

equivariantly, where

acts on the domain and codomain by

In particular,

acts on the fiber

. Now

is physically rigid if and only if

acts
transitively on

. Therefore

. But

, it follows that

which gives the desired inequality!
¡õ
04/01/2014
Perverse sheaves
Let

be a field. For a separated finite type

-scheme

, we have a triangulated category

(the
bounded derived category of
constructible 
-adic sheaves on

) equipped with a standard

-structure such that there is an equivalence of categories

For

a morphism, we have adjoint pairs

and

. We also have adjoint pairs

for

.
The triangulated category
is defined to be the colimit of
. The latter triangulated category
is hard to define (it is not defined as the derived category of
-sheaves, which do not have enough injectives). There are 3 approaches to define
..
- Use the pro-etale topology introduced by Bhatt-Scholze (
becomes a genuine sheaf).
- Taking limit is well-behaved for the stable
-category version of
. The triangulated limit comes for free.
- Deligne's classical approach: replace
with the full subcategory of very well-behaved complexed (these are quasi-isomorphic to bounded complexes of constructible
-flat sheaves). Call this full subcategory
. Then
is naturally triangulated:
is a distinguished triangle if
is a distinguished triangle for any
.

is
semi-perverse if for any

,

;

is
perverse if

and

are both semi-perverse, where

is the Verdier dual of

.
Suppose

is smooth of dimension

. For

lisse

-sheaf on

. Then
![$\mathcal{F}[d]$](./latex/latex2png-PureMotives_257849238_-5.gif)
is perverse (since
$](./latex/latex2png-PureMotives_137722692_-5.gif)
) but not for other shifts. In general, perverse sheaves are built out of lisse sheaves on smooth varieties.
Introducing perverse sheaves allows one to define intersection cohomology
for singular proper varieties
satisfying the Poincare duality and purity. Another major motivation for us is the following function-sheaf dictionary.
Take

and a

-sheaf

on

. We define for any

,

For example, when

,

produces the trace of the Frobenii on the cohomology of the fibers of the morphism

. Generalizing this, for any

, we define
The key thing is that these functions interact nicely with the sheaf-theoretic operations. For example,
- When
is a distinguished triangle, we have
.
.
- For
, we have
for
.
- For
, we have
(think:
is the integration over the fibers) This is essentially the Lefschetz trace formula.
The moral is that if you have some classically understood operations on functions, you can mimic them at the level of sheaves. The key role of perverse sheaves that one can recover the perverse sheaves from their functions:
Suppose

and

are two semisimple perverse sheaves. Then

and

are isomorphic if and only if

for any

.
How do we produce more perverse sheaves from the "lisse on smooth" case (Example 25)?
- Suppose
is an affine morphism, the
preserves semi-perversity (but not perversity).
- Suppose
is a quasi-finite morphism, then
preserves semi-perversity (but not perversity).
If

is both affine and quasi-finite (e.g.,

is an affine immersion), then both

and

preserve perversity.
Suppose

is perverse, then

is semi-perverse (by the previous theorem). Now

(by duality). Since

is perverse,

is perverse (by definition), hence

is also semi-perverse (by the previous theorem).
¡õ
Here comes the key construction: intermediate extensions. Let
be a locally closed immersion. For simplicity, let us assume that
is affine, so
is affine and quasi-finite. If
. Then both
and
lie in
. There is a natural map
.
Define the
intermediate extension (or
middle extension)

(in the abelian category

).
is fully faithful.
.
preserves simple objects, injections and surjections.
Any simple perverse sheaf

on

is of the form
![$j_{!*}(\mathbb{F}[\dim Y])$](./latex/latex2png-PureMotives_62835776_-5.gif)
for some smooth affine

locally closed subvariety of

, for some lisse sheaf

on

.
(Sketch)
Define

to be the closure of

. Choose

such that the constructible sheaves

become lisse when restricted to

. Take

. This works.
¡õ
Interesting things happen when extending
to the boundary of
.
04/08/2014
The middle convolution
Today we will introduce the key operation on perverse sheaves in Katz's classification of rigid local systems: the middle convolution.
The rigid local system considered in Example
23 is the sheaf of the local solutions of the Gauss hypergeometric equation. The solution has an integral representation

Here the parameter

determines the local monodromies. More generally,

is the solution of

This integral looks like

namely the (additive) convolution of

and

. The function

corresponds to the rank 1 Kummer sheaf associated to the representation

. Similarly, the function

corresponds to a tensor product of (translated) Kummer sheaves. So rigid local system

can be expressed in terms of the
convolution of simpler objects.
Here is the precise construction of the convolution.
Let

be an algebraically closed field. Let

be a connected smooth affine algebraic group. Let

be the multiplication map. For

, we can define two kinds of
convolutions 
and

.
Suppose

such that for all

, both

and

are perverse. We define the
middle convolution 
to be the image of

in the abelian category of perverse sheaves.
Take

and

. Let

be the local system on

associated to a nontrivial character

. Let
![$K=j_*\mathcal{L}_\chi[ 1 ]$](./latex/latex2png-PureMotives_94689304_-5.gif)
, where

. Then the middle convolution

makes sense: both

,

preserve perversity, by the following proposition.
Suppose

. Let

be irreducible such that its isomorphism class is not translation invariant. Then

and

both preserve perversity.
- The
statement follows from the
statement: because
is also perverse and not translation invariant, so
is perverse; taking dual implies that
is perverse.
- For
, then
is perverse if and only if
is semi-perverse:
is semi-perverse since
is affine.
- If
is perverse, then the followings are equivalent:
is perverse for any
;
is perverse for any irreducible
.
In fact, because
is an abelian category with all objects having finite length, we can induct on the length of
. A distinguished triangle
(with
lower lengths) gives a distinguished triangle
; the long exact sequence in cohomology then implies that 
- So we reduce to the case of irreducible perverse sheaves
. We now use the assumption that
. Namely, we need to check that
By Example 26, an irreducible perverse sheaf is either punctual or an intermediate extension
. If either
or
is punctual, then
is a translate of
or
, hence is perverse. So we can assume that there exists
and lisse
on
such that
and
. The stalk of
at a geometric point
is
This vanishes for
since
. It remains to check that for
, this vanishes for at most finitely many
. Now we need to use the assumption that
is not translation invariant. Notice the fiber
, so for
, we have
Since both
and
are lisse on
, it is equal to
Since both source and target are irreducible, this is zero unless there is an isomorphism
. Since the right hand side does not depend on
, either we win or there exists infinitely many
such that there is such an isomorphism. Since these
lie in the support of a constructible sheaf on a curve, the same would happen for
in an open dense subset
. Let
, then the isomorphism class of
is translation invariant under
. Thus we obtain a subgroup containing
, which must be the whole group, under which
is translation invariant. A contradiction!
¡õ
Henceforth we take
.
Now we can state the main results (slightly specialized) about the middle involution.
04/10/2014
Let us explain some of the ideas of the proof without going into details.
For any algebraic closed field
and any
separated and of finite type, we define the subcategory of middle extensions
consisting of
, where
is lisse for some
. We have an operation ![$$\otimes_\mathrm{mid}: \mathrm{ME}(X)\times \mathrm{ME}(X)\rightarrow\mathrm{ME}(X),\quad (j_{!*}\mathcal{F}[d]), j_{!*}\mathcal{G}[d])\mapsto j_{!*}(\mathcal{F}\otimes \mathcal{G})[d].$$](./latex/latex2png-PureMotives_253320503_.gif)
If
has characteristic
and
. It turns out that the category
on
satisfying
is equivalent to
via the Fourier transform. The middle convolution on
then corresponds to
on
(as in the classical Fourier theory: the Fourier transform of the convolution is the product of the Fourier transforms).
We now define the Fourier transform, which is a functor

. Fix an additive character

and denote the associated the Artin-Schrier sheaf on

by

. Let

be the two projections

. Motivated by the classical Fourier transform

we define
![$$\mathrm{FT}_{\psi,!}(\mathcal{F}):=R_{p_2,!}(p_1^*\mathcal{F}\otimes \mathcal{L}_{\psi(xy)}[ 1 ]),$$](./latex/latex2png-PureMotives_85805534_.gif)
where

is the pullback of

via the multiplication map. Similarly define

using

. It turns out that

and we denote it by

for short. It follows that

preserves

since projection maps are affine and the duality switches

and

. Moreover,

is
involutive:
![$$\mathrm{FT}_\psi\circ\mathrm{FT}_\psi\cong[-1]^*(-1).$$](./latex/latex2png-PureMotives_164867710_.gif)
In particular,

is an auto-equivalence of

.
Katz's classification
Using Theorem 21, we can prove the Katz's classification theorem of tamely ramified cohomological rigid local systems. Besides
, we also need a simpler twisting operation: If
is rank 1 lisse on
, we define
The index of rigidity is easily seen to be preserved under
.
Given an irreducible tame cohomological rigid local system on
, our aim is to apply a series of
and
's (these are all invertible operations) to obtain a rank 1 object (which is easy to understand).
(Katz)
Suppose

. Assume

is lisse on

(so

) and cohomologically rigid. Then there exists a generic rank 1

lisse on

and a nontrivial character

such that

has
strictly smaller rank than

.
Also, we can arrange
and
to have local monodromies contained in the local monodromies of
. So the local monodromies of
is contained in the local monodromies of
. In particular, if all th eigenvalues of the local monodromies of
are roots of unity, then we can arrange the same for
.
For

, we write

Similarly, at

, we write

For

, we write

to be the number of Jordan blocks of length

in the unipotent matrix

(which can be viewed as the dual partition of

).
To guess what to do, we look at the rank formula (proved via Fourier transform by Katz): for any nontrivial
,
If we want to drop the rank, we want to maximize the number of eigenvalues 1 by twisting and then take the middle convolution with respect the
that maximizes
.
For each
, we choose
such that
is maximal. We form
of rank 1 such that
, i.e.,
Then
has larger
than
for any
.
We replace
by the resulting twist
and choose
such that
is maximal. In order to apply Katz's rank formula, we claim that any such
is nontrivial. Assume this claim is true. By the rank formula and the Euler characteristic formula, we have
So to see the rank actually drops, we just need to show that
For this we need the cohomological rigidity
The last term can be written as
By the maximality of
, we know the last term is at most
So the cohomological rigidity implies that
which implies what we wanted since
where
by definition.
It remains to prove the claim. Assume that
is trivial, then the same argument implies that
But since
is irreducible, we know that
, a contradiction.
¡õ
04/15/2014
Since
and
are both reversible operations,
Given a tuple of monodromies (at each

), we can apply Katz's algorithem above to determine whether this tuple actually
arises as local monodromies of an irreducible cohomologically rigid local system. This solves the Deligne-Simpson problem in the cohomologically rigid irreducible case.
Starting with the above local monodromies, apply Katz's algorithem to reduce to the rank 1 case.
Local systems of type 
Let
be an algebraically closed field of characteristic not
(or 2).
- Fix
. Let
such that
. Then there exists an irreducible cohomologically rigid local system
of rank 7 with the local monodromies:
- at
:
,
- at
:
,
- at
: any of the following (determined by the conditions on
and
),
,
,
,
,
.
In each case, the monodromy group (the Zariski closure of the image of the monodromy representation) is
.
- Let
be cohomologically rigid, ramified at
and have monodromy group
. Then
is ramified at exactly two points
of
. Moreover, up to permuting
,
is conjugate to one of the local systems above.
We start with the second part. Suppose

is lisse exactly on

, we want to know how big

is. Since

is cohomologically rigid, we have

We look at the table of the centralizers of conjugacy classes of

in

(copied from Dettweiler-Reiter), we see the largest dimension is 29.

So
, hence
. When
, then the four centralizer adds to dimension 100. We look at the table again and there are only the following three cases
,
,
.
We can rule out all these three cases due to the necessary criterion for irreducibility:
. Namely, the Euler characteristic formula tells us that
or
For example,
case is ruled out because then the Jordan form is
by the table: if we twist by the character with local monodromies
at the three finite points (and
at
), we then get a new local system with local monodromies
, which violates the above irreducible criterion. Other cases are similar.
Now
. Again the cohomological rigidity implies that the sum of dimensions of the three centralizers is 51. The table implies that the possibilities are
,
,
,
,
,
,
.
The necessary criterion of irreducibility implies that
. This excludes the cases a), d), e), g). Since the monodromy group is assumed to be all of
(so far we only used the monodromy group is contained in
), the (14 dimensional) adjoint representation of
is also irreducible (the adjoint representation is irreducible for general simple reductive groups). Applying the irreducibility criterion gives
So the sum of dimensions of the three centralizers in
is less than 14. This excludes the cases b), f). The corresponding
centralizers for case c) must be
. The local monodromies are
at
,
at
,
- several possibilities at
.
The first part the follows by checking which of these possibilities can arise using Katz's algorithm.
For example, take the
case at the third point
. Write
for the rank 1 local system with local monodromies
at
and
at
. Twisting by
we get
,
and
. Then the rank formula together with the table tells us that the rank of the middle convolution
is equal to
, with local monodromies
,
,
. Twisting by
and take middle convolution, we obtain local monodromies
,
,
(rank 5),... until we get down to
,
,
(rank 2) and
(rank 1). The last rank 1 local system actually exists since
. Now running the algorithm reversely proves that the original local system also exists.
The final thing to do is to prove the monodromy group is actually
. First notice that our monodromy representation
is orthogonal. The dual representation has the same local monodromies (up to
-conjugacy); since
is physically rigid, this implies that
. Since the dimension 7 is odd,
must be orthogonal. So
maps into
.
We use the fact that an irreducible subgroup
of
lies inside an
-conjugate of
if and only if
. So we need to show that
, i.e.,
. By the Poincare duality, this is equivalent to
. We compute the Euler characteristic, 
- At
: the
is the image of
of
in
via the 6-th symmetric power
Since
is the same as the number of irreducible constituents of this
-representation, we know
.
- At
: the local monodromy is semisimple and it is easy to see that the dimension
(either two
's or no
's).
- At
: the local monodromy is the image of
of
via
. The number of irreducible constituents of
is 13.
So the Euler characteristic is
. Hence
. Therefore our
lands inside
. The fact (going back to Dynkin) is that an irreducible subgroup of
containing a regular unipotent element must be either
or
. The
rules out the possibility of
.
¡õ
04/17/2014
Universal rigid local systems
Let
be an algebraically closed field,
. Fix an order of quasi-unipotence
(
) and fix
a primitive
-th root of unity.
Let
and
be a cohomologically rigid local system, lisse on
with eigenvalues contained in
. We will show how to produce a local system over the arithmetic configuration space of
points, whose geometric fibers over
are cohomologically rigid objects in
lisse away from
and one of which gives the original
.
Let
![$R_{N,\ell}=\mathbb{Z}[\zeta_N,1/N \ell]$](./latex/latex2png-PureMotives_260666949_-5.gif)
and fix a nonzero map

. The configuration space over

is defined to be (

of)
The universal rigid local system will live on
, i.e.,
Notice that one can specialize to
via
.
04/22/2014
Middle convolution with parameters
Let
be a normal domain, finite type over
. Let
be a divisor given by the equation
, where
are all distinct. Let
be another divisor given by
.
The middle convolution is an operation
where
is given by
. For the application,
since
.
Define
Then we have the following diagram
where
are the projections and
is the difference morphism.
We denote
. Then the compactified projection
is proper smooth.
Let

,

, we define the
middle convolution 
and the
naive convolution
(Ideas of the proof)
a. Let us take
. Then
is the complement of hyperplanes in the
and
directions and the diagonal. Recall that
is the sheaf associated to
. Since the projection
is trivialized with respect to the stratification
, it can be computed as
. It follows that the middle convolution is lisse. For a formal proof, see Katz, Sommes exponentielles, Section 4.7.
b. Recall that if
is an
-sheaf on
(a scheme of finite type over
), we say that
is pure of weight
if for any closed points
,
is pure of weight
in the familiar sense. We say
is mixed if it admits a filtration by subsheaves such that
are all pure.
(Weil II)
Suppose

is a morphism of schemes of finite type over

(

is invertible on

). If

is mixed of weight

on

then

is mixed of weight

.
It follows that the naive convolution is mixed of weight
. The purity of the middle convolution follows from the analogous statement for curves over finite fields:
If

is a smooth curve,

lisse on

pure of weight

, then

is pure of weight

.
This result is less surprising by noticing that
(the source is mixed of weight
and the target is mixed of weight
). To show the graded piece statement, we use the short exact sequence of sheaves
which gives a long exact sequence
The last term is zero since
is punctual. Since
is an open immersion, we know that
has weights
by 1.8.9 in Weil II. So we are done since
-term is mixed of weight
.
c. We prove the ! version. We can check on geometric fibers and it suffices to show for
. Notice that
is lisse on
and
is the coinvariants under the local monodromy at
. In our case,
is unramified along
, but
is indeed ramified, and hence the coinvariants is 0.
¡õ
Suppose

is affine open,

is separated finite type of an algebraically closed field

.

is proper and

is finite. Suppose

such that

and

are perverse. Then

in

.
We have two exact sequences in

,

The kernel and cokernel are supported on

, applying

, we obtain two distinguished triangle on

, four out of the six terms are perverse by assumption. One can check that

is perverse by taking the long exact sequence on cohomology, hence the two distinguished triangles are indeed short exact sequences of perverse sheaves. Splicing together we obtain the result.
¡õ