Variation of local fundamental group and local first homology group

Abstract

In this paper, we discuss various results about variation of the local fundamental group of normal complex spaces. It is proved that the finite Galois descent of upper semicontinuity of the local fundamental group holds at a factorial complex analytic germ. We also show by an example that finite Galois descent of upper semicontinuity of the local first homology group at a smooth germ is not true in general. We prove that the local fundamental groups of a normal complex algebraic variety are finite in number upto isomorphism.