In this paper, we prove a general theorem concerning the analyticity of the closure of a subspace defined by a family of variations of mixed Hodge structures, which includes the analyticity of the zero loci of degenerating normal functions. For the proof, we use a moduli of the valuative version of log mixed Hodge structures.
"Analyticity of the closures of some Hodge theoretic subspaces." Proc. Japan Acad. Ser. A Math. Sci. 87 (9) 167 - 172, November 2011. https://doi.org/10.3792/pjaa.87.167