We start by recalling the main theorem of this seminar (Weil II for curves) and illustrating some of its arithmetic consequences. Then we introduce the theory of Lefschetz pencils and deduce the last part (Riemann hypothesis) of the Weil conjectures. Finally, we prove the geometric semisimplicity of lisse pure sheaves, and use it to deduce the hard Lefschetz theorem.
This is an expanded note prepared for a STAGE talk, Spring 2014. Our main references are [1], [2] and [3].
Weil II for curves
Recall that we have proved the following target theorem.
Let

be a smooth compactification over

. Then

is
-pure of weight

for

.
: By Leray or Gysin, we have
, which consists of the
-invariants, is
-pure of weight
. (Note: cohomology groups are always understood as taking the cohomology of the base change
).
: By Leray or Gysin, we have
. Since
is
-mixed of weight
, and by Poincare duality,
is
-mixed of weight
, it follows
is
-pure of weight
.
: by birational invariance of top
(or long exact sequence associated to
), we have
, which is
-pure of weight
.
¡õ
Several remarkable arithmetic consequences follows.
(Riemann Hypothesis over finite fields)
Let

be a smooth projective geometrically connected variety. Then for any

and any

,

is

-pure of weight

.
We will deduce this theorem in the next section. The Riemann hypothesis illustrates the following surprising slogan: the arithmetic of a smooth projective variety

over a finite field

is controlled by the topology of the corresponding complex manifold

. If we factorize the zeta function of

as

here

(resp.

) are the

eigenvalues on the odd (resp. even) degree cohomology groups

, then by the Lefschetz trace formula,

The Riemann hypothesis provides the key to understand these

and

: it allows us to obtain very good estimates of

as long as we know enough about the cohomology of

. For example, when

is an elliptic curve over

, we obtain

By Corollary
1, we know that

, as

eigenvalues on

, has pure weight 1. In this way we recover the classical Hasse-Weil bound,
(Ramanujan conjecture)
The Ramanujan

-function is defined to be the coefficients of the

-expansion of the weight 12 cusp eigenform

Ramanujan famously observed (!) without proof that

. This turns out to be a general phenomenon for coefficients of a cusp eigenform

of weight

and level

, as a consequence of Weil II together with the algebro-geometric incarnation of cusp eigenforms. Let

be the modular curves over
![$\mathbb{Z}[1/N]$](./latex/latex2png-WeilII_136422478_-5.gif)
with the universal family of elliptic curves

. By the Eichler-Shimura relation, the Hecke eigenvalue

(

) of

matches up with the trace of

on the (2-dimensional)

-isotypic component of

Since

is lisse on

of pure of weight 1, by Corollary
1, the above

is pure of weight

. Therefore both

eigenvalues have absolute values

and so

.
Notice in this example, it is convenient to work with non-constant coefficient systems. Here is another typical example.
(Kloosterman sum)
Let

. The classical Kloosterman sum is defined to be

This can be interpreted geometrically as follows. Let

be the Artin-Schrier sheaf on

associated to the additive character

Let

be the smooth affine variety of dimension

defined by the equation

and define

Then

Deligne computed that

Hence by the strong version of Weil II, we obtain the estimate

, which is certainly not easy to obtain using elementary methods.
Lefschetz pencils and Riemann Hypothesis
When
is a curve, the Riemann hypothesis follows from Corollary 1, since
is
-pure of weight 0 for any
. For the general case, we induct on
via the theory of Lefschetz pencils.

Using incidence correspondences and the Bertini theorem, one can show the existence of Lefschetz pencils.
There exists a Lefschetz pencil of hyperplanes defined over

on

, after possibly a finite extension of the base field and possibly replacing the projective embedding from

by

for some

.
Let
be the blow up of
along
, then we obtain a projective morphism
with smooth fibers over
, where
is a finite set of points. After possibly a finite base extension, we may assume
consists of
-rational points. The Leray spectral sequence for the blow up
implies that
So it suffices to prove the purity statement for
.

Due to the simple nature of singularities, it is possible to describe both the local and global monodromy actions on the cohomology. In the complex setting, this is classically known as the Picard-Lefschetz theory. In the
-adic setting, this is done in SGA 7 and is briefly summarized as follows.
Now we can finish the induction step. For simplicity let us assume
is odd (the even case is similar). Let
and
. Then by the property of the Lefschetz pencil (Theorem 3 a), b)), we have
. The Leray spectral sequence implies that
By induction and proper base change,
is lisse on
, pure of weight
. So Corollary 1 implies that each term
is pure of weight
. Hence
is pure of weight
as well. This completes the proof of Theorem 2.
The hard Lefschetz theorem
(Geometric semisimplicity)
Let

be a finite field and

be a prime number. Let

be a smooth geometrically connected variety. Let

be a lisse

-pure

-sheaf on

. Then the representation

of

associated to

is semisimple.
As we have already seen in Koji's talk, we can replace

by a space filling curve without changing the geometric monodromy group, the smallest algebraic group containing the image of

. So it suffices to treat the curve case. We are going to induct on the length of

as a

-representation. When

is irreducible as a

-representation, as in Kestutis's talk,

is semisimple as a

-representation (since

is normal). Now suppose

is an extension of lisse

-pure sheaves on

,

we would like to show that there is a section (as

-representations). In other words, we would like to show that the element

corresponding to the identity morphism

lies in the image of the first map in the following sequence,

The crucial thing is the
mismatch of weights in the second map: by Theorem
1, the source has weight 0 but the target has weight 1. Since

is fixed by

, it must die in

and hence comes from some element of

.
¡õ
Next we will see how the geometric semisimplicity grew out of the arithmetic consideration of weights can help us to understand the fundamental geometric structure of smooth projective varieties.
(Hard Lefschetz)
Let

be a projective smooth connected variety over an algebraically closed field (of any characteristic). Let

be an ample line bundle on

and

. Then for any

, the

-th iterated cup product

is an isomorphism.
Since

and

are defined over a finitely generated subfield

of

, using the defining equations, we obtain a morphism of scheme

and an ample line bundle on

with the generic fiber

and

, where

is scheme of a finite type over

. After possibly shrinking

, we may assume

has projective smooth connected fibers. To show

is an isomorphism on over generic point, it suffices to check it is an isomorphism on each closed point. This puts us in the situation where

is the algebraic closure of a
finite field (even if we work with

at the beginning).
The case
is trivial. We are going to induct on the dimension of
. Take a Lefschetz pencil
on
. Let
be a smooth hyperplane section. By the Lefschetz hyperplane theorem,
is an isomorphism when
and is an injection when
(I am going to omit all the Tate twists due to my laziness). Taking Poincare dual,
is an isomorphism when
and is surjective when
. Now by the projection formula,
can be decomposed as
When
,
and
are isomorphisms. By induction hypothesis,
is an isomorphism and it follows that
is also an isomorphism. It remains to treat the key case
. In this case,
is an injection and
is a surjection. By Poincare duality,
is an isomorphism is equivalent to that the pairing
is non-degenerate. Using the injection
, it is equivalent to that the pairing
is non-degenerate on the image of
. By the property of the Lefschetz pencil (Theorem 3 c)), we have
Now we use Theorem 5: the
-action on
is semisimple (this is the only place we use Weil II). So we obtain a
-equivariant decomposition
for some
without trivial
-constituents. Hence the non-degenerate cup product pairing on
decomposes accordingly and in particular restricts to a non-degenerate pairing on
, as desired.
¡õ
We mention one immediate geometric consequence of the hard Lefschetz theorem to end this talk.
The

-th Betti number

is even for odd

.
The hard Lefschetz together with the Poincare duality provides a non-degenerate pairing on

, which is
alternating when

is odd.
¡õ
References
[1]N. Katz, L-functions and monodromy: four lectures on Weil II, 2000, http://web.math.princeton.edu/~nmk/arizona34.pdf.
[2]Milne, James S., Lectures on Etale Cohomology (v2.10), Available at www.jmilne.org/math/.
[3]Deligne, Pierre, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no.52, 137--252.