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.


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