Geodesic
Homology reduction of cycles in the complement of an algebraic hypersurface
Sbornik. Mathematics, Tome 186 (1995) no. 10, pp. 1417-1427 
N. A. Buruchenko; A. K. Tsikh. Homology reduction of cycles in the complement of an algebraic hypersurface. Sbornik. Mathematics, Tome 186 (1995) no. 10, pp. 1417-1427. http://geodesic.mathdoc.fr/item/SM_1995_186_10_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

An example of a 3-dimensional cycle in the complement of an algebraic hypersurface $V\subset\mathbb C^3$ that cannot be deformed into a tube over (is not homologous to the coboundary of) a 2-dimensional cycle in the set of regular points of $V$ is presented. Thus, the corresponding result of Poincare in $\mathbb C^2$ fails in $\mathbb C^n$ for $n>2$. It is proved that Poincare's result holds for hypersurfaces in $\mathbb C^n$ with a 'thin' set of singularities that are complete intersections.

[1] Poincare H., “Sur les residus des integrales doubles”, Acta Math., 90:3 (1887), 460–541 | MR

[2] Lere Zh., Differentsialnoe i integralnoe ischislenie na kompleksnom analiticheskom mnogoobrazii, IL, M., 1961

[3] Aizenberg L. A., Yuzhakov A. P., Integralnye predstavleniya i vychety v mnogomernom kompleksnom analize, Nauka, Novosibirsk, 1979

[4] Griffiths P. A., “On the periods of certain rational integrals”, Ann. Math., 90:3 (1969), 460–514 | DOI | MR

[5] Gordon G., “The residues calculus in several complex variables”, Trans. AMS, 213 (1975), 127–176 | DOI | MR | Zbl

[6] Dolbo D., “Obschaya teoriya mnogomernykh vychetov”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 7, VINITI, M., 1985, 227–251 | MR

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

[8] Poly J. B., “Sur un theoreme J. Leray en theorie des residus”, C.R.A.S. Paris, 274:2 (1972), 171–174 | MR | Zbl