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

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.