Vasudevan Srinivas

On finite presentation for the tame fundamental group

This is a report on joint work with H. Esnault and M. Schusterman.

Recall that the etale 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 etale 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 fundamental
groups of smooth projective varieties.