Programming and Computer Software

, Volume 42, Issue 2, pp 99–106 | Cite as

Algorithm for construction of volume forms on toric varieties starting from a convex integer polytope

  • A. A. KytmanovEmail author
  • A. V. Shchuplev
  • T. V. Zykova


This paper presents a method and a corresponding algorithm for constructing volume forms (and related forms that act as kernels of integral representations) on toric varieties from a convex integer polytope. The algorithm is implemented in the Maple computer algebra system. The constructed volume forms are similar to the volume forms of the Fubini–Study metric on a complex projective space and can be used for constructing integral representations of holomorphic functions in polycircular regions of a multidimensional complex space.


Projective Space Volume Form Toric Variety Cauchy Kernel Empty List 
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  1. 1.
    Audin, M., The Topology of Torus Actions on Symplectic Manifolds, Basel: Birkhauser, 1991.CrossRefzbMATHGoogle Scholar
  2. 2.
    Cox, D.A., The homogeneous coordinate ring of a toric variety, J. Alg. Geom, 1995, no. 4, pp. 17–50.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Fulton, W., Introduction to Toric Varieties: Annals of Mathematics Studies, Princeton: Princeton Univ. Press, 1993.Google Scholar
  4. 4.
    Kytmanov, A.A., An analog of the Fubini–Studi form for two-dimensional toric varieties, Sib. Math. J., 2003, vol. 44, no. 2. pp. 286–297.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kytmanov, A.A. and Semusheva, A.Y., Averaging of the Cauchy kernels and integral realization of the local residue, Mathematische Zeitschrift, 2010, vol. 264, no. 1. pp. 87–98.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Shchuplev, A.V., Tsikh, A.K., and Yger, A., Residual kernels with singularities on coordinate planes, Proc. Steklov Inst. Math., 2006, vol. 253, no. 1. pp. 256–274.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Shabat, B.V., Raspredelenie znachenii golomorfnykh otobrazhenii (Distribution of Values of Holomorphic Mappings), Moscow: Nauka, 1982.zbMATHGoogle Scholar
  8. 8.
    Kytmanov, A.A., Algorithm for constructing an integral representation from the fan of a toric variety, J. Math. Sci., 2012, vol. 186, no. 3. pp. 453–460.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Batyrev, V.V., Quantum cohomology ring of toric manifolds, Journees de Geometrie Algebrique d’Orsay, 1993, no. 218, pp. 9–34.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kytmanov, A.A. and Shchuplev, A.V., An algorithm for constructing toric compactifications, Program. Comput. Software, 2013, vol. 39, no. 4. pp. 207–211.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Passare, M., Amoebas, convexity, and the volume of integer polytopes, Adv. Stud. Pure Math., 2004, no. 42, pp. 263–268.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Griffiths, P. and Harris, J., Principles of Algebraic Geometry, New York: Wiley, 1978.zbMATHGoogle Scholar
  13. 13.
    Kytmanov, A.M., Integral Bokhnera–Martinelli i ego primeneniya (Bochner–Martinelli Integral and Its Applications), Novosibirsk: Nauka, 1992.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • A. A. Kytmanov
    • 1
    Email author
  • A. V. Shchuplev
    • 1
  • T. V. Zykova
    • 1
  1. 1.Siberian Federal UniversityKrasnoyarskRussia

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