  We show that the zero locus of a normal function on a smooth complex
algebraic variety S is algebraic provided that the normal function extends to a
admissible normal function on a smooth compactification of S with torsion
singularity. This result generalizes our previous result for admissible normal
functions on curves [arxiv:math/0604345 [math.AG]]. It has also been obtained
by M. Saito using a different method in a recent preprint [arXiv:0803.2771v2].
