Last update: 06 July 2012
Let be a projective space of dimension on a nonsingular projective variety, a linear subspace of codimension two, the dual line, an algebraic closure of and on obtained from by extension of scalars. The diagram [De; 5.1.1] of [De; 5.6] provides an analogue of the diagram over
Suppose is connected of even dimension and the pencil of hyperplane sections of defined by is a Lefschetz pencil. The set of such that is singular is defined over i.e. providing We set and
Let The vanishing part of the cohomology is stable under thus defines over a local system of This last object is defined over is the reciprocal image of the sheaf on and, on is the reciprocal image of a local system
The cup product is an alternating form Denoting the orthogonal complement to relative to by in we see that induces a perfect duality
For every the polynomial has rational coefficients.
Let be the inclusion of in and that of in The eigenvalues of acting on are algebraic numbers such that all their complex conjugates satisfy
Let be a locally constant sheaf on such that its reciprocal image on is a constant sheaf. Then there exist units in such that for every we have
This lemma expresses as the reciprocal image of a sheaf on knowing its direct image on This last is identified with an adic representation of and we take
The Lemma 1.3 [De; 6.4] applies to to and to
For the fibre is a variety over the finite field If is a point of over is deduced from by extension of scalars of to its algebraic closure and is the fibre of at The formula [De; 1.5.4] for the variety over is thus written and is the product of with
Putting and applying Lemma 1.3 [De; 6.4] to the factors and we find that there exist adic units and in such that for every and, in particular, the second factor is in If coincides with it is reasonable to simultaneously remove this from the list of and this from the list of the Thus we may assume, and we will assume, that for all and all
It suffices to prove that the polynomials and have rational coefficients, i.e. that the family of (resp. the family of ) is defined over We will deduce the following propositions.
Let and be two families of adic units in We assume that If is a very large finite set of integers and is a very large subset of of density 0, then, if satisfies (for every ) and the denominator of written in irreducible form, is
Let be a finite set of integers and and two families of elements of a field. If, for very large, not divisible by any of the the family of coincides with that of the (up to rearrangement), then the family of coincides with the family of (up to rearrangement).
We proceed by induction on The set of integers such that is an ideal We prove that there exists such that If not the will be different from 1, and there exist arbitrarily large integers not divisible by any of the nor by any of the We will have and this will contradict the hypothesis. Thus there exists such that We conclude by applying the induction hypothesis to the families and
Let and be two families of adic units in and We assume that for every divides Then divides
Disregarding the pairs of common elements in the families and verify the hypothesis of Proposition 2.1 [De; 6.6]. Apply Proposition 2.1 [De; 6.6]. By hypothesis, the rational fractions (W6 6.6.1) are polynomials. Thus no remains, which means that divides
This proposition furnishes an intrinsic characterisation of in terms of the family of polynomials It is the lcm of the polynomials which satisfies the hypothesis of Proposition 2.3 [De; 6.8].
We prove [De; 6.5] and thus Theorem 1.1 [De; 6.2] (modulo Proposition 2.1 [De; 6.6]). Take and in Proposition 2.1 [De; 6.6]. We find an intrinsic characterization of the family of in terms of the family of rational fractions Since these are in the family of is defined over
The polynomials are thus in Proposition 2.3 [De; 6.8] provides an intrinsic description of the family of in terms of this family of polynomials. The family of is thus defined over
Preliminaries: Let and be the fibre of at The arithmetic fundamental group extension of (generator: ) by the geometric fundamental group acts on by symplectic similitudes: We denote by the multiplier of a symplectic similitude Let be the subgroup defined by the equation ( being an adic unit, is defined for all ). The fact that has values in can be expressed by saying that the map from into with coordinates the canonical projection on and factors
The image of is open in
In fact, projects onto and the image of in is open [De; 5.10].
For an adic unit, the set of such that is an eigenvalue of is closed of measure 0.
It is clear that is closed. For each let be the set of such that and let be the set of such that is an eigenvalue of Then is a homogeneous space for and we check that is a proper algebraic subspace, thus of measure 0. By Lemma 4.1 [De; 6.11], is thus of measure 0 in the inverse image of in and we apply Fubini to the projection
We prove Proposition 2.1 [De; 6.6]. For each and the set of integers such that is the set of multiples of a fixed integer (we don't exclude ). By hypothesis,
By Lemma 4.2 [De; 6.12] and the density theorem of Čebotarev, the set of such that a is an eigenvalue of acting on is of density 0. We take for the set of and for the set of as above.
[De] P. Deligne, La conjecture de Weil: I, Publications mathématiques de l'I.H.É.S., tome 43, (1974), p. 273-307.