### On finite presentation for the tame fundamental group

Heilbronn Number Theory Seminar

8th June 2022, 2:30 pm – 3:30 pm

Fry Building, 2.04

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.

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