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In this paper, the notion of local algebraic fundamental groups of normalcomplex analytic singularities are generalized to certain profinite groupscalled $D$-local algebraic fundamental groups which turns out to be useful evenfor the study of usual local algebraic fundamental groups and the Lefshetz typetheorem for $D$-local algebraic fundamental groups is proved under certainconditions. The theorem yields, for example, the finiteness of the localalgebraic fundamental groups of a certain class of four dimensionalsingularities and will be useful for the classification of three dimensionalpurely log terminal singularities.

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