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

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

. 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

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

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

,

(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

.

¡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

, 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

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

is perverse (since

) 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

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

, 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

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

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*:

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

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.

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