Space of nilpotent orbits and extension of period maps (I): The weight 3 Calabi-Yau types

In Kato-Nakayama-Usui's theory, a certain space of nilpotent orbits can be constructed and serve as a completion of a given period map. This can be regarded as a generalization of Mumford's toroidal compactification for locally symmetric varieties. Kato-Nakayama-Usui's construction requires the existence of a weak fan, which is not known in general for non-classical cases. In this paper, we show after some slight modifications, such weak fans exist for a large class of period maps of weight 3 Calabi-Yau type. In particular, for these cases a Kato-Nakayama-Usui type completion can be constructed.