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MOTIVICITY OF THE MIXED HODGE STRUCTURE OF SOME DEGENERATIONS OF CURVES

  • Chae, Hi-Joon ;
  • Jun, Byung-Heup
  • Received : 2008.12.19
  • Published : 2010.05.31

Abstract

We consider a degeneration of genus 2 curves, which is opposite to maximal degeneration in a sense. Such a degeneration of curves yields a variation of mixed Hodge structure with monodromy weight filtration. The mixed Hodge structure at each fibre, which is different from the limit mixed Hodge structure of Schmid and Steenbrink, can be realized as $H^1$ of a noncompact singular elliptic curve. We also prove that the pull back of the above variation of mixed Hodge structure to a double cover of the base space comes from a family of noncompact singular elliptic curves.

Keywords

variation of mixed Hodge structure

References

  1. J. Carlson, The geometry of the extension class of a mixed Hodge structure, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 199-222, Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI, 1987.
  2. J. Carlson, S. Muller-Stach, and C. Peters, Period Mappings and Period Domains, Cambridge Studies in Advanced Mathematics, 85. Cambridge University Press, Cambridge, 2003.
  3. P. Deligne, Theorie de Hodge. II, Inst. Hautes Etudes Sci. Publ. Math. No. 40 (1971), 5-57.
  4. P. Deligne, Theorie de Hodge. III, Inst. Hautes Etudes Sci. Publ. Math. No. 44 (1974), 5-77.
  5. P. Deligne, La conjecture de Weil. II, Inst. Hautes Etudes Sci. Publ. Math. No. 52 (1980), 137-252.
  6. P. Deligne, Local behavior of Hodge structures at infinity, Mirror symmetry, II, 683-699, AMS/IP Stud. Adv. Math., 1, Amer. Math. Soc., Providence, RI, 1997.
  7. P. Griffiths and L. Tu, Asymptotic behavior of a variation of Hodge structure, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), 63-74, Ann. of Math. Stud., 106, Princeton Univ. Press, Princeton, NJ, 1984.
  8. P. Griffiths and J. Harris, Princeples of Algebraic Geometry, Wiley, New York, 1978.
  9. L. Illusie, Autour du theoreme de monodromie locale, Asterisque Vol. 223 (1994), 9-58.
  10. M. Kontsevich and D. Zagier, Periods, Mathematics unlimited-2001 and beyond, 771-808, Springer, Berlin, 2001.
  11. W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211-319. https://doi.org/10.1007/BF01389674
  12. J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986.
  13. J. Steenbrink, Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229-257. https://doi.org/10.1007/BF01403146
  14. J. Steenbrink and S. Zucker, Variation of mixed Hodge structure. I, Invent. Math. 80 (1985), no. 3, 489-542. https://doi.org/10.1007/BF01388729
  15. S. Muller-Stach, A remark on height pairings, Algebraic cycles and Hodge theory (Torino, 1993), 253-259, Lecture Notes in Math., 1594, Springer, Berlin, 1994.