Finite generation of nilpotent quotients of fundamental groups of punctured spectra

TL;DR

This paper proves that the maximal pro-nilpotent quotient of the étale fundamental group of punctured spectra of certain local domains is topologically finitely generated, using advanced algebraic geometry techniques.

Contribution

It establishes a weaker form of Grothendieck's conjecture by showing finite generation of the pro-nilpotent quotient, expanding understanding of fundamental groups in algebraic geometry.

Findings

Maximal pro-nilpotent quotients are topologically finitely generated.

Uses $p$-adic nearby cycles and intersection pairing analysis.

Provides new insights into deformation cohomology and algebraic group structures.

Abstract

In SGA 2, Grothendieck conjectures that the \'etale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In this paper, we prove a weaker statement, namely that the maximal pro-nilpotent quotient of the fundamental group is topologically finitely generated. The proof uses $p$-adic nearby cycles and negative definiteness of intersection pairings over resolutions of singularities as well as some analysis of Lie algebras of certain algebraic group structures on deformation cohomology.