Images of extended period maps
Author:
Sampei Usui
Journal:
J. Algebraic Geom. 15 (2006), 603621
DOI:
https://doi.org/10.1090/S1056391106004504
Published electronically:
June 20, 2006
MathSciNet review:
2237263
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Abstract  References  Additional Information
Abstract: As a geometric application of polarized log Hodge structures, we show the following. Let $M_{H}^{\mathrm {sm}}$ be a projective variety which is a compactification of the coarse moduli space of surfaces of general type constructed by Kawamata, Kollár, ShepherdBarron, 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 $M_{H}^{\mathrm {sm}}$, 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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Additional Information
Sampei Usui
Affiliation:
Graduate School of Science, Osaka University, Toyonaka, Osaka, 5600043, Japan
Email:
usui@math.sci.osakau.ac.jp
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 GrantsinAid for Scientific Research (B) No. 15340009, the Ministry of Education, Science, Sports and Culture, Japan