Following C. Simpson, we show that every variation of graded-polarized mixed Hodge structure defined over Q carries a natural Higgs bundle structure∂ + θ which is invariant under the C * action studied in [20]. We then specialize our construction to the context of [6], and show that the resulting Higgs field θ determines (and is determined by) the Gromov-Witten potential of the underlying family of Calabi-Yau threefolds.

We prove an analog of Schmid's SL 2 -orbit theorem for a class of variations of mixed Hodge structure which includes logarithmic deformations, degenerations of 1-motives and archimedean heights. In particular, as consequence this theorem, we obtain a simple formula for the asymptotic behavior of the archimedean height of a flat family of algebraic cycles which depends only on the weight filtration and local monodromy.

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic.In Part 2 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 We then use the answer to recover an unpublished theorem of Deligne, which characterizes the sl 2 -splitting of a real mixed Hodge structure. Corollary 1.3. If S is algebraic then the zero locus of an admissible normal function ν : S → J(H) is an algebraic subvariety of S.

We study certain spaces of nilpotent orbits in the Hodge domains introduced by [GGK1], and treat a number of examples. More precisely, we compute the Mumford-Tate group of the limit mixed Hodge structure of a generic such orbit. The result is used to present these spaces as iteratively fibered algebraic-group orbits in a minimal way. 1 arXiv:1210.5301v2 [math.AG] 8 Apr 2015 9 These are stated for the rank-one case, cf. §2 (where B(N ) and the {I p,q } are defined); they have obvious generalizations replacing N with σ. 10 In [GGK1] (cf. Example VI.B.15), it was discovered that (the finitely many) connected components of the subset of a period domain comprising Hodge structures with Mumford-Tate group contained in a given (Mumford-Tate) subgroup of Aut(V, Q), need not have the same generic Mumford-Tate group.

We study certain real Lie-group orbits in the compact duals of Mumford-Tate domains, verifying a prediction of [GGK1] and determining which orbits contain a limit point of some period map. A variety of examples are worked out for the groups SU (2, 1), Sp 4 , and G 2 .