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POLARIZED REAL TORI

  • Received : 2014.01.23
  • Published : 2015.03.01

Abstract

For a fixed positive integer g, we let $\mathcal{P}_g=\{Y{\in}\mathbb{R}^{(g,g)}{\mid}Y=^tY>0\}$ be the open convex cone in the Euclidean space $\mathbb{R}^{g(g+1)/2}$. Then the general linear group GL(g, $\mathbb{R}$) acts naturally on $\mathcal{P}_g$ by $A{\star}Y=AY^tA(A{\in}GL(g,\mathbb{R}),\;Y{\in}\mathcal{P}_g)$. We introduce a notion of polarized real tori. We show that the open cone $\mathcal{P}_g$ parametrizes principally polarized real tori of dimension g and that the Minkowski modular space 𝔗g = $GL(g,\mathbb{Z}){\backslash}\mathcal{P}_g$ may be regarded as a moduli space of principally polarized real tori of dimension g. We also study smooth line bundles on a polarized real torus by relating them to holomorphic line bundles on its associated polarized real abelian variety.

Keywords

References

  1. A. A. Albert, Symmetric and alternate matrices in an arbitrary field. I, Trans. Amer. Math. Soc. 43 (1938), no. 3, 386-436. https://doi.org/10.1090/S0002-9947-1938-1501952-6
  2. A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Lie Groups: History, Frontiers and Applications, Vol. IV, Math. Sci. Press, Brookline, Mass., 1975.
  3. W. Baily, Satake's compactification of $V^*_n $, Amer. J. Math. 80 (1958), 348-364. https://doi.org/10.2307/2372789
  4. W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math. 84 (1966), 442-528. https://doi.org/10.2307/1970457
  5. C. Birkenhake and H. Lange, Complex Tori, Progress in Mathematics, 177. Birkhauser Boston, Inc., Boston, 1999.
  6. H. Comessati, Sulle varieta abeliane reali. I, II, Ann. Mat. Pura. Appl. 2 (1924), 67-106
  7. H. Comessati, Sulle varieta abeliane reali. I, II, Ann. Mat. Pura. Appl 4 (1926), 27-72.
  8. G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Ergebnisse der Math. 22, Springer-Verlag, Berlin-Heidelberg-New York, 1990.
  9. M. Goresky and Y. S. Tai, The moduli space of real abelian varieties with level structure, Compositio Math. 139 (2003), no. 1, 1-27. https://doi.org/10.1023/B:COMP.0000005079.56232.e3
  10. S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984.
  11. J. Igusa, Theta Functions, Springer-Verlag, Berlin-Heidelberg-New York, 1972.
  12. M. Itoh, On the Yang Problem (SFT), preprint, Max-Planck Institut fur Mathematik, Bonn, 2011.
  13. A.W. Knapp, Representation Theory of Semisimple Groups, Princeton University Press, Princeton, New Jersey, 1986.
  14. H. Lange and C. Birkenhake, Complex Abelian Varieties, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1992.
  15. H. Maass, Siegel modular forms and Dirichlet series, Lecture Notes in Math. 216, Springer-Verlag, Berlin-Heidelberg-New York, 1971.
  16. Y. Matsushima, On the intermediate cohomology group of a holomorphic line bundle over a complex torus, Osaka J. Math. 16 (1979), no. 3, 617-631.
  17. H. Minkowski, Gesammelte Abhandlungen, Chelsea, New York, 1967.
  18. D. Mumford, Abelian Varieties, Oxford University Press, 1970; Reprinted, 1985.
  19. I. Satake, On the compactification of the Siegel space, J. Indian Math. Soc. 20 (1956), 259-281.
  20. I. Satake, Algebraic Structures of Symmetric Domains, Kano Memorial Lectures 4, Iwanami Shoton, Publishers and Princeton University Press, 1980.
  21. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemann- ian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87.
  22. M. Seppala and R. Silhol, Moduli spaces for real algebraic curves and real abelian vari- eties, Math. Z. 201 (1989), no. 2, 151-165. https://doi.org/10.1007/BF01160673
  23. G. Shimura, On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II (1975), no. 17, 261-268.
  24. C. L. Siegel, Symplectic geometry, Amer. J. Math. 65 (1943), 1-86 https://doi.org/10.2307/2371774
  25. C. L. Siegel, Symplectic geometry, Academic Press, New York and London, 1964
  26. Gesammelte Abhandlungen, no. 41, vol. II, 274-359, Springer-Verlag, 1966,
  27. R. Silhol, Real Abelian varieties and the theory of Comessatti, Math. Z. 181 (1982), no. 3, 345-364. https://doi.org/10.1007/BF01161982
  28. R. Silhol,, Real Algebraic Surfaces, Lecture Notes in Math. 1392, Springer-Verlag, Berlin- Heidelberg-New York, 1989.
  29. R. Silhol,, Compactifications of moduli spaces in real algebraic geometry, Invent. Math. 107 (1992), no. 1, 151-202. https://doi.org/10.1007/BF01231886
  30. J.-H. Yang, A note on holomorphic vector bundles over complex tori, Bull. Korean Math. Soc. 23 (1986), no. 2, 149-154.
  31. J.-H. Yang, Holomorphic vector bundles over complex tori, J. Korean Math. Soc. 26 (1989), no. 1, 117-142.
  32. J.-H. Yang, A note on a fundamental domain for Siegel-Jacobi space, Houston J. Math. 32 (2006), no. 3, 701-712.
  33. J.-H. Yang, Invariant metrics and Laplacians on Siegel-Jacobi space, J. Number Theory 127 (2007), no. 1, 83-102. https://doi.org/10.1016/j.jnt.2006.12.014
  34. J.-H. Yang, A partial Cayley transform of Siegel-Jacobi disk, J. Korean Math. Soc. 45 (2008), no. 3, 781-794. https://doi.org/10.4134/JKMS.2008.45.3.781
  35. J.-H. Yang, Remark on harmonic analysis on the Siegel-Jacobi space, arXiv:1107.0509v1 [math.NT], 2009.
  36. J.-H. Yang, Invariant metrics and Laplacians on Siegel-Jacobi disk, Chin. Ann. Math. Ser. B 31 (2010), no. 1, 85-100. https://doi.org/10.1007/s11401-008-0348-7
  37. J.-H. Yang, Invariant differential operators on the Siegel-Jacobi space and Maass-Jacobi forms, Proceedings of the International Conference on Geometry, Number Theory and Representation Theory, 39-63, KM Kyung Moon Sa, Seoul, 2013.
  38. J.-H. Yang, Invariant differential operators on the Minkowski-Euclid space, J. KoreanMath. Soc. 50 (2013), no. 2, 275-306.