de Mathématiques de Luminy
Ghorpade Sudhir R., Lachaud
Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields.
We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field.This extends a result of Deligne for nonsingular complete intersections.For normal complete intersections,this inequality generalizes also the classical Lang-Weil inequality. Moreover,we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein.We also prove a conjecture of Lang and Weil concerning the Picard varieties and étale cohomology spaces of projective varieties.The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck-Lefschetz Trace Formula.We also describe some auxiliary results concerningth étale cohomology spaces and Betti numbers of projective varieties over flnite fields and a conjecture alongwith some partial results concerning the number of points of projective algebraic sets over finite fields.