Extension of period maps by polyhedral fans

Kato and Usui developed a theory of partial compactifications for quotients of period domains D by arithmetic groups Γ, in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases. Their partial compactifications, which aim to fully compactify the im...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2022-09, Vol.406, p.108532, Article 108532
Main Author: Deng, Haohua
Format: Article
Language:English
Subjects:
Complete fan
Hodge theory
Kato-Usui theory
Nilpotent orbit
Period map
Toroidal compactification
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Summary:Kato and Usui developed a theory of partial compactifications for quotients of period domains D by arithmetic groups Γ, in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases. Their partial compactifications, which aim to fully compactify the images of period maps, rely on the choice of a fan which is strongly compatible with Γ. In particular, they conjectured the existence of a complete fan, which would serve to simultaneously compactify all period maps of a given type. In this article, we briefly review the theory, and construct a fan which compactifies the image of a period map arising from a particular two-parameter family of Calabi-Yau threefolds studied by Hosono and Takagi, with Hodge numbers (1,2,2,1). On the other hand, we disprove the existence of complete fans in some general cases, including the (1,2,2,1) case.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2022.108532