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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral
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by Toshiyuki Mano and Humihiko Watanabe PDF
Proc. Amer. Math. Soc. 140 (2012), 3867-3881 Request permission

Abstract:

We study twisted cohomology and homology groups on a one-dimensional complex torus minus $n$ distinct points with coefficients in a certain local system of rank one. This local system comes from the integrand of the Riemann-Wirtinger integral introduced by Mano. We construct bases of non-vanishing cohomology and homology groups, give an interpretation as a pairing of a cohomology class and a homology class to the Riemann-Wirtinger integral, and finally describe briefly the Gauss-Manin connection on the cohomology groups.
References
  • Kazuhiko Aomoto, Les équations aux différences linéaires et les intégrales des fonctions multiformes, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 271–297 (French). MR 508176
  • Kazuhiko Aomoto, On the structure of integrals of power product of linear functions, Sci. Papers College Gen. Ed. Univ. Tokyo 27 (1977), no. 2, 49–61. MR 590052
  • Kazuhiko Aomoto, Une correction et un complément à l’article: “Les équations aux différences linéaires et les intégrales des fonctions multiformes” [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 271–297; MR 58 #22688c], J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), no. 3, 519–523 (French). MR 560011
  • Kazuhiko Aomoto, Configurations and invariant Gauss-Manin connections of integrals. I, Tokyo J. Math. 5 (1982), no. 2, 249–287. MR 688126, DOI 10.3836/tjm/1270214894
  • Kazuhiko Aomoto, Configurations and invariant Gauss-Manin connections for integrals. II, Tokyo J. Math. 6 (1983), no. 1, 1–24. MR 707836, DOI 10.3836/tjm/1270214323
  • Kazuhiko Aomoto and Michitake Kita, Theory of hypergeometric functions, Springer Monographs in Mathematics, Springer-Verlag, Tokyo, 2011. With an appendix by Toshitake Kohno; Translated from the Japanese by Kenji Iohara. MR 2799182, DOI 10.1007/978-4-431-53938-4
  • K. Chandrasekharan, Elliptic functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396, DOI 10.1007/978-3-642-52244-4
  • Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • Giovanni Felder and Alexander Varchenko, Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, Internat. Math. Res. Notices 5 (1995), 221–233. MR 1333749, DOI 10.1155/S1073792895000171
  • Otto Forster, Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81, Springer-Verlag, New York-Berlin, 1981. Translated from the German by Bruce Gilligan. MR 648106
  • R. C. Gunning, Lectures on Riemann surfaces, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966. MR 0207977
  • Ko-Ki Ito, The elliptic hypergeometric functions associated to the configuration space of points on an elliptic curve. I. Twisted cycles, J. Math. Kyoto Univ. 49 (2009), no. 4, 719–733. MR 2591113, DOI 10.1215/kjm/1265899479
  • Nicholas M. Katz and Tadao Oda, On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 (1968), 199–213. MR 237510, DOI 10.1215/kjm/1250524135
  • Toshiyuki Mano, Studies on monodromy preserving deformation of linear differential equations on elliptic curves, J. Math. Phys. 50 (2009), no. 10, 103501, 21. MR 2573133, DOI 10.1063/1.3204973
  • Toshiyuki Mano, The Riemann-Wirtinger integral and monodromy-preserving deformation on elliptic curves, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn110, 19. MR 2448089, DOI 10.1093/imrn/rnn110
  • Riemann, B., Gesammelte mathematische Werke. In Nachträge, edited by M. Noether and W. Wirtinger. Leipzig, Germany: Teubner, 1902.
  • Humihiko Watanabe, Transformation relations of matrix functions associated to the hypergeometric function of Gauss under modular transformations, J. Math. Soc. Japan 59 (2007), no. 1, 113–126. MR 2302665
  • Humihiko Watanabe, Twisted homology and cohomology groups associated to the Wirtinger integral, J. Math. Soc. Japan 59 (2007), no. 4, 1067–1080. MR 2370006, DOI 10.2969/jmsj/05941067
  • Humihiko Watanabe, Linear differential relations satisfied by Wirtinger integrals, Hokkaido Math. J. 38 (2009), no. 1, 83–95. MR 2501895, DOI 10.14492/hokmj/1248787013
  • Watanabe, H., On the general transformation of the Wirtinger integral, preprint.
  • Wirtinger, W., Zur Darstellung der hypergeometrischen Function durch bestimmte Integrale, Akad. Wiss. Wien. S.-B. IIa, 111 (1902), 894-900.
  • Wirtinger, W., Eine neue Verallgemeinerung der hypergeometrischen Integrale, Akad. Wiss. Wien. S.-B. IIa, 112 (1903), 1721-1733.
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Additional Information
  • Toshiyuki Mano
  • Affiliation: Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara-cho, Okinawa 903-0213, Japan
  • Email: tmano@math.u-ryukyu.ac.jp
  • Humihiko Watanabe
  • Affiliation: Kitami Institute of Technology, 165, Koencho, Kitami 090-8507, Hokkaido, Japan
  • Email: hwatanab@cs.kitami-it.ac.jp
  • Received by editor(s): December 7, 2009
  • Received by editor(s) in revised form: August 21, 2010, January 31, 2011, April 19, 2011, and April 28, 2011
  • Published electronically: March 8, 2012
  • Additional Notes: The first author was supported in part by GCOE, Kyoto University and MEXT Grant-in-Aid for Young Scientists (B) (No. 21740118).
    The second author was supported in part by Grant-in-Aid for Scientific Research (C) (No. 19540158), JSPS

  • Dedicated: Dedicated to Professor Keizo Yamaguchi on his sixtieth birthday
  • Communicated by: Ted Chinburg
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 3867-3881
  • MSC (2010): Primary 33C05; Secondary 14K25, 55N25, 14F40, 32C35
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11221-3
  • MathSciNet review: 2944728