Journal of Algebraic Geometry Journal of Algebraic Geometry

  Previous issue | Table of contents | Next issue
Recently posted articles | Most recent issue | All issues
     

Images of extended period maps

Author(s): Sampei Usui
Journal: J. Algebraic Geom. 15 (2006), 603-621.
Posted: June 20, 2006
Retrieve article in: PDF DVI PostScript

Abstract | References | Additional information

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.

References:

[AK]
D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0 (eprint), alg-geom/9707012.

[Al]
V. Alexeev, Boundedness and $ K^{2}$ for log surfaces, Internat. J. Math. 5 (1994), 779-810. MR 1298994 (95k:14048)

[AM]
V. Alexeev and S. Mori, Bounding singular surfaces of general type, in `` Algebra, Arithmetic and Geometry with Applications" (Christensen et al., ed.), Springer-Verlag, 2003, pp. 143-174. MR 2037085 (2005f:14077)

[Ar]
M. Artin, Algebraization of formal moduli II: Existence of modifications, Ann. Math. (2) 91 (1970), 88-135. MR 0260747 (41:5370)

[F]
T. Fujisawa, Limits of Hodge structures in several variables, Compositio Math. 115 (1999), 129-183. MR 1668986 (99m:14019)

[GR]
H. Grauert, R. Remmert, Coherent analytic sheaves, Grund. Math. Wiss. 265, Springer-Verlag, 1984. MR 0755331 (86a:32001)

[I]
L. Illusie, Logarithmic spaces $ ($according to K. Kato$ )$, in: Barsotti Symposium in Algebraic Geometry (V. Critstante and W. Messing, eds.), Perspectives in Math. 15, Academic Press, 1994, pp. 183-203. MR 1307397 (95j:14023)

[IKN]
L. Illusie, K. Kato and C. Nakayama, Quasi-unipotent logarithmic Riemann-Hilbert correspondences, J. Math. Sci. Univ. Tokyo 12 (2005), no. 1, 1-66. MR 2126784 (2006a:14030)

[Kar]
K. Karu, Minimal models and boundedness of stable varieties, J. Algebraic Geom. 9 (2000), 93-109. MR 1713521 (2001g:14059)

[Kf1]
F. Kato, Log smooth deformation theory, Tôhoku Math. J. 48 (1996), 317-354. MR 1404507 (99a:14012)

[Kf2]
-, The relative log Poincaré lemma and relative log de Rham theory, Duke Math. J. 93-1 (1998), 179-206. MR 1620096 (99g:32015)

[Kk1]
K. Kato, Logarithmic structures of Fontaine-Illusie, in `` Algebraic analysis, geometry, and number theory" (J.-I. Igusa, ed.), Perspectives in Math., Johns Hopkins University Press, Baltimore, 1989, pp. 191-224. MR 1463703 (99b:14020)

[Kk2]
-, Toric singularity, Amer. J. Math. 116 (1994), 1073-1099. MR 1296725 (95g:14056)

[KkNc]
K. Kato and C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $ \mathbf C$, Kodai Math. J. 22 (1999), 161-186. MR 1700591 (2000i:14023)

[KMN]
K. Kato, T. Matsubara and C. Nakayama, Log $ C^{\infty }$-functions and degenerations of Hodge structures, Advanced Studies in Pure Math. 36: Algebraic Geometry 2000, Azumino, (2002), 269-320. MR 1971519 (2004i:32023)

[KyNy]
Y. Kawamata and Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), 395-409. MR 1296351 (95j:32030)

[KU1]
K. Kato and S. Usui, Logarithmic Hodge structures and classifying spaces (summary), in CRM Proc. & Lect. Notes: The Arithmetic and Geometry of Algebraic Cycles (NATO Advanced Study Institute / CRM Summer School 1998: Banff, Canada) 24 (1999), 115-130. MR 1736878 (2001e:14009)

[KU2]
-, Borel-Serre spaces and spaces of SL(2)-orbits, Advanced Studies in Pure Math. 36: Algebraic Geometry 2000, Azumino, (2002), 321-382. MR 1971520 (2004f:14021)

[KU3]
-, Classifying spaces of degenerating polarized Hodge structures, preprint (submitted).

[Kaw]
Y. Kawamata, Crepant blowing-ups of $ 3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. Math. 127 (1988), 93-163. MR 0924674 (89d:14023)

[Ko1]
J. Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), 235-268. MR 1064874 (92e:14008)

[Ko2]
-, Log surfaces of general type; Some conjectures, Contemporary Math., Amer. Math. Soc., Providence, RI 162 (1994), 261-275. MR 1272703 (95c:14042)

[KSB]
J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299-338. MR 0922803 (88m:14022)

[Mb1]
T. Matsubara, On log Hodge structures of higher direct images, Kodai Math. J. 21 (1998), 81-101. MR 1645599 (2000e:32028)

[Mb2]
-, Log Hodge structures of higher direct images in several variables, preprint.

[Ms]
T. Matsusaka, On polarized normal varieties, I, Nagoya Math. J. 104 (1986), 175-211. MR 0868444 (88e:14011)

[Mo]
B. Moishezon, The algebraic analog of compact complex spaces with a sufficiently large field of meromorphic functions, Izv. Akad. Nauk SSSR Ser. Mat. (translation: Math. Ussr-Izv.) 33 (1969), I: 174-238, II: 323-367, III: 506-548. MR 0260748 (41:5371)

[N]
C. Nakayama, A projection formula for log smooth varieties in log étale cohomology, TITECH MATH 12-97 (64).

[R]
M. Reid, The moduli space of 3-folds with $ K=0$ may nevertheless be irreducible, Math. Ann. 278 (1987), 329-334. MR 0909231 (88h:32016)

[U1]
S. Usui, Recovery of vanishing cycles by log geometry, Tôhoku Math. J. 53-1 (2001), 1-36. MR 1808639 (2002a:32010)

[U2]
-, Recovery of vanishing cycles by log geometry: Case of several variables, in Commutative Algebra and Algebraic Geometry, and Computational Methods: Proc. Internat. Conference, Hanoi, 1996 (D. Eisenbud, ed.), Springer-Verlag, 1999, pp. 133-143. MR 1714854 (2000g:14014)

[V]
E. Viehweg, Quasi-projective moduli for polarized manifolds, Springer-Verlag, 1995. MR 1368632 (97j:14001)

Additional Information:

Sampei Usui
Affiliation: Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
Email: usui@math.sci.osaka-u.ac.jp

PII: S 1056-3911(06)00450-4
Received by editor(s): July 22, 2004
Received by editor(s) in revised form: April 7, 2005
Posted: 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

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2007 University Press, Inc.
Comments: jag-query@ams.org
AMS Website