Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Images of extended period maps

Author: Sampei Usui
Journal: J. Algebraic Geom. 15 (2006), 603-621
Published electronically: June 20, 2006
MathSciNet review: 2237263
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Abstract: As a geometric application of polarized log Hodge structures, we show the following. Let $ M_{H}^{{sm}}$ be a projective variety which is a compactification of the coarse moduli space of surfaces of general type constructed by Kawamata, Kollár, Shepherd-Barron, Alexeev, Mori, Karu, et al., and let $ \Gamma\backslash D_{\Sigma}$ be a log manifold which is the fine moduli space of polarized log Hodge structures constructed by Kato and Usui. If we take a suitable finite cover $ M'\to M_{i}$ of any irreducible component $ M_{i}$ of $ M_{H}^{{sm}}$, and if we assume the existence of a suitable fan $ \Sigma$, then there is an extended period map $ \psi :M'\to \Gamma \backslash D_{\Sigma}$ and its image is the analytic subspace associated to a separated compact algebraic space. The point is that, although $ \Gamma \backslash D_{\Sigma}$ is a `` log manifold" with slits, the image $ \psi (M')$ is not affected by these slits and is a classical familiar object: a separated compact algebraic space.

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Sampei Usui
Affiliation: Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

Received by editor(s): July 22, 2004
Received by editor(s) in revised form: April 7, 2005
Published electronically: June 20, 2006
Additional Notes: Partly supported by the Grants-in-Aid for Scientific Research (B) No. 15340009, the Ministry of Education, Science, Sports and Culture, Japan

American Mathematical Society