$D$-local algebraic fundamental groups of germs of normal complex analytic singularities

TL;DR

This paper introduces $D$-local algebraic fundamental groups for normal complex analytic singularities, extending existing concepts and proving a Lefschetz type theorem that aids in classifying complex singularities.

Contribution

It generalizes local algebraic fundamental groups to $D$-local groups and establishes a Lefschetz type theorem for them, advancing singularity classification.

Findings

Proves finiteness of local algebraic fundamental groups for certain 4D singularities

Establishes a Lefschetz type theorem for $D$-local algebraic fundamental groups

Provides tools for classifying 3D purely log terminal singularities

Abstract

In this paper, the notion of local algebraic fundamental groups of normal complex analytic singularities are generalized to certain profinite groups called $D$-local algebraic fundamental groups which turns out to be useful even for the study of usual local algebraic fundamental groups and the Lefshetz type theorem for $D$-local algebraic fundamental groups is proved under certain conditions. The theorem yields, for example, the finiteness of the local algebraic fundamental groups of a certain class of four dimensional singularities and will be useful for the classification of three dimensional purely log terminal singularities.