Geodesic
Residual Kernels with Singularities on Coordinate Planes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 277-295.

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A finite collection of planes $\{E_\nu \}$ in $\mathbb C^d$ is called an atomic family if the top de Rham cohomology group of its complement is generated by a single element. A closed differential form generating this group is called a residual kernel for the atomic family. We construct new residual kernels in the case when $E_\nu$ are coordinate planes such that the complement $\mathbb C^d\setminus \bigcup E_\nu$ admits a toric action with the orbit space being homeomorphic to a compact projective toric variety. They generalize the well-known Bochner–Martinelli and Sorani differential forms. The kernels obtained are used to establish a new formula of integral representations for functions holomorphic in Reinhardt polyhedra. 
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A. V. Shchuplev; A. K. Tsikh; A. Yger. Residual Kernels with Singularities on Coordinate Planes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 277-295. http://geodesic.mathdoc.fr/item/TM_2006_253_a20/

[1] Aizenberg L.A., Yuzhakov A.P., Integral representations and residues in multidimensional complex analysis, Transl. Math. Monogr., 58, Amer. Math. Soc., Providence (RI), 1983 | MR | Zbl

[2] Aleksandrov P.S., “Topological duality theorems. I: Closed sets”, AMS Transl. Ser. 2, 30 (1963), 1–102 | MR | MR | Zbl

[3] Batyrev V.V., “Quantum cohomology ring of toric manifolds”, Journées de géométrie algébrique d'Orsay (Orsay, 1992), Astérisque, 218, Soc. math. France, Paris, 1993, 9–34 | MR | Zbl

[4] Cox D., “The homogeneous coordinate ring of a toric variety”, J. Alg. Geom., 4 (1995), 17–50 | MR | Zbl

[5] Cox D., “Toric residues”, Ark. Mat., 34:1 (1996), 73–96 | DOI | MR | Zbl

[6] Cox D., “Recent developments in toric geometry”, Algebraic geometry (Santa Cruz, 1995), Proc. Symp. Pure Math., 62, pt. 2, Amer. Math. Soc., Providence (RI), 1997, 389–436 | MR | Zbl

[7] Deligne P., Goresky M., MacPherson R., “L'algèbre de cohomologie du complément, dans un espace affine, d'une famille finie de sous-espaces affines”, Michigan Math. J., 48 (2000), 121–136 | DOI | MR | Zbl

[8] Ermolaeva T.O., Tsikh A.K., “Integrirovanie ratsionalnykh funktsii po $\mathbb R^n$ s pomoschyu toricheskikh kompaktifikatsii i mnogomernykh vychetov”, Mat. sb., 187:9 (1996), 45–64 | MR | Zbl

[9] Fulton W., Introduction to toric varieties, Ann. Math. Stud., 131, Princeton Univ. Press, Princeton (NJ), 1993 | MR | Zbl

[10] Goresky M., MacPherson R., Stratified Morse theory, Ergeb. Math. und Grenzgeb. 3. Flg., 14, Springer, Berlin, 1988 | MR | Zbl

[11] Griffiths Ph., Harris J., Principles of algebraic geometry, J. Wiley Sons, New York, 1978 | MR | Zbl

[12] Harvey F.R., “Integral formulae connected by Dolbeault's isomorphism”, Rice Univ. Stud., 56:2 (1970), 77–97 | MR

[13] Khovanskii A.G., “Algebra and mixed volumes”, Geometric inequalities, Addendum 3, Grundl. Math. Wissensch., 285, eds. Burago Yu.D., Zalgaller V.A., Springer, Berlin, 1988

[14] Kirwan F.C., Cohomology of quotients in symplectic and algebraic geometry, Math. Notes, 31, Princeton Univ. Press, Princeton (NJ), 1984 | MR | Zbl

[15] Koppelman W., “The Cauchy integral for differential forms”, Bull. Amer. Math. Soc., 73 (1967), 554–556 | DOI | MR | Zbl

[16] Kouchnirenko A.G., “Polyèdres de Newton et nombres de Milnor”, Invent. math., 32:1 (1976), 1–31 | DOI | MR

[17] Kytmanov A.A., “Ob analoge formy Fubini–Shtudi dlya dvumernykh toricheskikh mnogoobrazii”, Sib. mat. zhurn., 44:2 (2003), 358–371 | MR | Zbl

[18] Leray J., “Le calcul différentiel et intégral sur une variété analytique complexe (Problème de Cauchy. III)”, Bull. Soc. math. France, 87 (1959), 81–180 | MR | Zbl

[19] Lieb I., Michel J., The Cauchy–Riemann complex. Integral formulae and Neumann problem, Aspects Math., E34, F. Vieweg Sohn, Braunschweig, 2002 | MR

[20] Kytmanov A.M., Integral Bokhnera–Martinelli i ego primeneniya, Novosibirsk, 1992 | Zbl

[21] Passare M., “Amoebas, convexity and the volume of integer polytopes”, Complex analysis in several variables, Adv. Stud. Pure Math., 42, Math. Soc. Japan, Tokyo, 2004, 263–268 | MR | Zbl

[22] Schuplev A.V., “O vosproizvodyaschikh yadrakh v $\mathbb C^d$ i formakh ob'ema na toricheskikh mnogoobraziyakh”, UMN, 60:2 (2005), 179–180 | MR

[23] Sorani G., “Integral representations of holomorphic functions”, Amer. J. Math., 88 (1966), 737–746 | DOI | MR | Zbl

[24] Tsikh A.K., Mnogomernye vychety i ikh primeneniya, Nauka, Novosibirsk, 1988 | MR | Zbl

[25] Tsikh A.K., “Toriska residyer”, Proc. Conf. on Complex Analysis “Nordan 3” (Stockholm, Apr. 22–25, 1999), Stockholm Univ., Stockholm, 1999, 16