François Charles

The Tate conjecture for K3 surfaces over finite fields

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.

The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K 3 surfaces

We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K 3 surface. The proof involves Verbitsky’s theoryxa0of hyperholomorphic sheaves and a studyxa0of the cohomology algebra of Hilbert schemes ofxa0 K 3 surfaces.

Conjugate varieties with distinct real cohomology algebras

Abstract Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field K and two complex embeddings of K, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l-adic coefficients are canonically isomorphic for any prime number l, and answers a question of Grothendieck.

Bertini irreducibility theorems over finite fields

Given a geometrically irreducible subscheme X in P^n over F_q of dimension at least 2, we prove that the fraction of degree d hypersurfaces H such that the intersection of H and X is geometrically irreducible tends to 1 as d tends to infinity. We also prove variants in which X is over an extension of F_q, and in which the immersion of X in P^n is replaced by a more general morphism.

Remarks on the Lefschetz standard conjecture and hyperkähler varieties

We study the Lefschetz standard conjecture on a smooth complex projective variety X. In degree 2, we reduce it to a local statement concerning local deformations of vector bundles on X. When X is hyperkahler, we show that the existence of nontrivial deformations of stable hyperholomorphic bundles implies the Lefschetz standard conjecture in codimension 2.

La conjecture de Tate entière pour les cubiques de dimension quatre

We prove the integral Tate conjecture for cycles of codimension

Exceptional isogenies between reductions of pairs of elliptic curves

2

On the Picard number of K3 surfaces over number fields

n on smooth cubic fourfolds over an algebraic closure of a field finitely generated over its prime subfield and of characteristic different from

Birational boundedness for holomorphic symplectic varieties, Zarhin’s trick for

2

K3

n or