  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.
