On the algebraicity of the zero locus of an admissible normal function

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic. In Part II of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the sl2-splitting of a real mixed Hodge structure. © © The Author(s) 2013