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Averaging of the Cauchy kernels and integral realization of the local residue. (English) Zbl 1186.14053
It is known that Bochner-Martinelli integral formula can be obtained by averaging of the Cauchy formula on some positive measures. In this paper similar formulas for a family of kernels of integral representation associated with toric variety are obtained. These formulas was studied by the first author in some of his previous papers. Here the mentioned kernels generalize the considered integral forms. Applications are given for integral realization of the local residue in algebraic geometry. The paper is interesting for a broad circle of specialists.

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels) 32A27 Residues for several complex variables
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##### References:
 [1] Griffiths P., Harris J.: Principles of Algebraic Geometry, pp. 813. Wiley, New York (1978) · Zbl 0408.14001 [2] Kytmanov A.A.: An analog of the Fubini-Studi form for two-dimensional toric varieties. Sib. Math. J. 44(2), 286–297 (2003) · Zbl 1081.32004 · doi:10.1023/A:1022936921419 [3] Kytmanov A.A.: An analog of the Bochner-Martinelli representation in d-circular polyhedra in the space $${\mathbb{C}^d}$$ . Russ. Math. 49(3), 49–55 (2005) · Zbl 1117.32004 [4] Kytmanov A.A.: Integral representations and volume forms on Hirzebruch surfaces. J. Sib. Federal Univ. 2, 3–9 (2008) [5] Shaimkulov B.A., Tsikh A.K.: Integral realizations of Grothendieck residue and its transformation under compositions (Russian). Vestnik KrasGU Fiz. Mat. Nauki Krasnoyarsk 1, 151–155 (2005) [6] Shchuplev A.V., Tsikh A.K, Yger A.: Residual kernels with singularities on coordinate planes. Proc. Steklov Inst. Math. 253, 256–274 (2006) · Zbl 1351.32010 · doi:10.1134/S0081543806020210 [7] Shchuplev, A.V.: Toric Varieties and Residues. Doctoral Thesis, Department of Math., Stockholm Univ., p. 70 (2007) [8] Tong T.L.: Integral representation formulae and Grothendieck residue symbol. Am. J. Math. 4, 904–917 (1973) · Zbl 0291.32008 · doi:10.2307/2373701 [9] Tsikh A., Yger A.: Residue currents. J. Math. Sci. New York 120(6), 1916–1971 (2004) · Zbl 1070.32003 · doi:10.1023/B:JOTH.0000020710.57247.b7
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