## On the fundamental group of a unirational variety. (Sur le groupe fondamental d’une variété unirationelle.)(French)Zbl 0591.14010

The author introduces new birational invariants for a nonsingular complete variety $$X$$ defined over a finite field $$\mathbb F_q$$, $$q=p^r$$, which are defined in terms of the étale and crystalline cohomologies of $$X$$. Then he applies these invariants to show that the fundamental group of a unirational, nonsingular, complete variety defined over an algebraically closed field of characteristic $$p>0$$ has a trivial $$p$$-Sylow group. If the variety is projective instead of being complete, the result is due to N. Suwa.

### MSC:

 14E20 Coverings in algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology 14M20 Rational and unirational varieties 14G15 Finite ground fields in algebraic geometry