Abstract
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.
References
Audin, M., The Topology of Torus Actions on Symplectic Manifolds, Basel: Birkhauser, 1991.
Cox, D.A., The homogeneous coordinate ring of a toric variety, J. Alg. Geom, 1995, no. 4, pp. 17–50.
Fulton, W., Introduction to Toric Varieties: Annals of Mathematics Studies, Princeton: Princeton Univ. Press, 1993.
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.
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.
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.
Shabat, B.V., Raspredelenie znachenii golomorfnykh otobrazhenii (Distribution of Values of Holomorphic Mappings), Moscow: Nauka, 1982.
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.
Batyrev, V.V., Quantum cohomology ring of toric manifolds, Journees de Geometrie Algebrique d’Orsay, 1993, no. 218, pp. 9–34.
Kytmanov, A.A. and Shchuplev, A.V., An algorithm for constructing toric compactifications, Program. Comput. Software, 2013, vol. 39, no. 4. pp. 207–211.
Passare, M., Amoebas, convexity, and the volume of integer polytopes, Adv. Stud. Pure Math., 2004, no. 42, pp. 263–268.
Griffiths, P. and Harris, J., Principles of Algebraic Geometry, New York: Wiley, 1978.
Kytmanov, A.M., Integral Bokhnera–Martinelli i ego primeneniya (Bochner–Martinelli Integral and Its Applications), Novosibirsk: Nauka, 1992.
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Original Russian Text © A.A. Kytmanov, A.V. Shchuplev, T.V. Zykova, 2016, published in Programmirovanie, 2016, Vol. 42, No. 2.
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Kytmanov, A.A., Shchuplev, A.V. & Zykova, T.V. Algorithm for construction of volume forms on toric varieties starting from a convex integer polytope. Program Comput Soft 42, 99–106 (2016). https://doi.org/10.1134/S0361768816020055
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DOI: https://doi.org/10.1134/S0361768816020055