Analyticity of the closures of some Hodge theoretic subspaces
  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. 
Kazuya Kato, Chikara Nakayama, Sampei Usui
math.AG