On finite presentation for the tame fundamental group

Abstract:

This is a report on joint work with H. Esnault and M. Schusterman. Recall that the étale fundamental group of a variety over an algebraically closed field of characteristic 0 is known to be a finitely presented profinite group; this is proved by first reducing to varieties over the complex numbers, and then comparing with the topological fundamental group. In positive characteristics, even if we restrict to smooth varieties, finite generation fails in general for étale fundamental groups of non-proper varieties (eg, for the affine line).

For a smooth variety with a smooth, projective compactification with a SNC boundary divisor, we show that the tame fundamental group is a finitely presented profinite group. In particular, this holds for the étale fundamental groups of smooth projective varieties.