Geodesic
Homology reduction of cycles in the complement of an algebraic hypersurface
Sbornik. Mathematics, Tome 186 (1995) no. 10, pp. 1417-1427 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_1995_186_10_a1,
     author = {N. A. Buruchenko and A. K. Tsikh},
     title = {Homology reduction of cycles in the~complement of an~algebraic hypersurface},
     journal = {Sbornik. Mathematics},
     pages = {1417--1427},
     year = {1995},
     volume = {186},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_10_a1/}
}
TY  - JOUR
AU  - N. A. Buruchenko
AU  - A. K. Tsikh
TI  - Homology reduction of cycles in the complement of an algebraic hypersurface
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 1417
EP  - 1427
VL  - 186
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_1995_186_10_a1/
LA  - en
ID  - SM_1995_186_10_a1
ER  - 
%0 Journal Article
%A N. A. Buruchenko
%A A. K. Tsikh
%T Homology reduction of cycles in the complement of an algebraic hypersurface
%J Sbornik. Mathematics
%D 1995
%P 1417-1427
%V 186
%N 10
%U http://geodesic.mathdoc.fr/item/SM_1995_186_10_a1/
%G en
%F SM_1995_186_10_a1
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/

[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