Extension of period maps by polyhedral fans

Kato and Usui developed a theory of partial compactifications for quotients period domains D by arithmetic groups Γ, in an attempt to generalize the toroidal Ash-Mumford-Rapoport-Tai non-classical cases. Their compactifications, which aim fully compactify images maps, rely on choice fan is strongly compatible with Γ. In particular, they conjectured existence complete fan, would serve simultaneously all maps given type. this article, we briefly review theory, construct compactifies image map arising from particular two-parameter family Calabi-Yau threefolds studied Hosono Takagi, Hodge numbers (1,2,2,1). On other hand, disprove fans some general cases, including (1,2,2,1) case.