On the weight filtration of the homology of algebraic varieties : the generalized Leray cycles
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 4, pp. 869-903.

Let Y be a normal crossing divisor in the smooth complex projective algebraic variety X and let U be a tubular neighbourhood of Y in X. Using geometrical properties of different intersections of the irreducible components of Y, and of the embedding YX, we provide the “normal forms” of a set of geometrical cycles which generate H * (A,B), where (A,B) is one of the following pairs (Y,), (X,Y), (X,X-Y), (X-Y,) and (U,). The construction is compatible with the weights in H * (A,B,) of Deligne’s mixed Hodge structure. The main technical part is to construct “the generalized Leray inverse image” of chains of the components of Y, giving rise to a chain situated in U.

Classification: 14C30, 14F25
@article{ASNSP_2002_5_1_4_869_0,
     author = {Elzein, Fouad and N\'emethi, Andr\'as},
     title = {On the weight filtration of the homology of algebraic varieties : the generalized {Leray} cycles},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {869--903},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {4},
     year = {2002},
     mrnumber = {1991006},
     zbl = {1098.14006},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2002_5_1_4_869_0/}
}
TY  - JOUR
AU  - Elzein, Fouad
AU  - Némethi, András
TI  - On the weight filtration of the homology of algebraic varieties : the generalized Leray cycles
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2002
SP  - 869
EP  - 903
VL  - 1
IS  - 4
PB  - Scuola normale superiore
UR  - http://archive.numdam.org/item/ASNSP_2002_5_1_4_869_0/
LA  - en
ID  - ASNSP_2002_5_1_4_869_0
ER  - 
%0 Journal Article
%A Elzein, Fouad
%A Némethi, András
%T On the weight filtration of the homology of algebraic varieties : the generalized Leray cycles
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2002
%P 869-903
%V 1
%N 4
%I Scuola normale superiore
%U http://archive.numdam.org/item/ASNSP_2002_5_1_4_869_0/
%G en
%F ASNSP_2002_5_1_4_869_0
Elzein, Fouad; Némethi, András. On the weight filtration of the homology of algebraic varieties : the generalized Leray cycles. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 4, pp. 869-903. http://archive.numdam.org/item/ASNSP_2002_5_1_4_869_0/

[1] N. A'Campo, La fonction zéta d'une monodromie, Commentarii Mathematici Helvetici 50 (1975), 233-248. | MR | Zbl

[2] A. Borel et al., “Seminar on Intersection Cohomology”, Forschungsinstitut für Mathematik ETH Zürich, 1984. | Zbl

[3] G. E. Bredon, Topology and Geometry, In: “Graduate Texts in Mathematics”, 139, Springer 1993. | MR | Zbl

[4] G. E. Bredon, Sheaf Theory, In: “Graduate Texts in Mathematics”, 170, Springer 1997. | MR | Zbl

[5] C. H. Clemens, Degeneration of Kähler manifolds, Duke Math. Journal 44 (1977), 215-290. | MR | Zbl

[6] P. Deligne, Théorie de Hodge, II, III, Publ. Math. IHES 40, 44 (1972, 1975), 5-47, 6-77. | Numdam | MR | Zbl

[7] F. El Zein, Introduction à la théorie de Hodge mixed, In: “Actualiés Mathématiques”, Hermann, Paris 1991. | MR | Zbl

[8] A. Fujiki, Duality of Mixed Hodge Structures of Algebraic Varieties, Publ. RIMS, Kyoto Univ., 16 (1980), 635-667. | MR | Zbl

[9] M. Goresky - R. Macpherson, Intersection Homology Theory, Topology, 19 (1980), 135-162. | MR | Zbl

[10] M. Goresky - R. Macpherson, Intersection Homology II, Invent. Math. 71 (1983), 77-129. | MR | Zbl

[11] P. Griffiths - W. Schmid, Recent developments in Hodge Theory, In: “Discrete subgroups of Lie groups”, Bombay Colloquium, Oxford Univ. Press, 1973. | Zbl

[12] F. Guillén - V. Navarro Aznar - P. Pascual-Gainza - F. Puerto, “Hyperrésolutions cubiques et descente cohomologique”, Lecture Notes in Math., 1335, Springer-Verlag 1988. | Zbl

[13] N. Habegger - L. Saper, Intersection cohomology of cs-spaces and Zeeman's filtration, Invent. Math. 105 (1991), 247-272. | MR | Zbl

[14] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79 (1964), 109-326. | MR | Zbl

[15] H. Hironaka, Triangulations of algebraic sets, Proceedings of Symposia in Pure Mathematics 29 (1975), 165-185. | MR | Zbl

[16] H. C. King, Topological invariance of intersection homology without sheaves, Topology and its applications 20 (1985), 149-160. | MR | Zbl

[17] J. Leray, Problème de Cauchy III, Bull. Soc. Math. France 87 (1959), 81-180. | Numdam | MR | Zbl

[18] R. Macpherson, “Intersection homology and perverse sheaves”, Lecture notes distributed at the 97th AMS meeting, San Francisco, 1991.

[19] C. Mccrory, On the topology of Deligne weight filtration, Proc. of Symp. in Pure Math. 40, Part 2 (1983), 217-226. | MR | Zbl

[20] V. Navarro Aznar, Sur la Théorie de Hodge des Variétés Algébriques à Singularités Isolées, Astérisque 130 (1985), 272-305. | Numdam | MR | Zbl

[21] A. Parusiński, “Blow-analytic retraction onto the central fibre”, Real analytic and algebraic singularities (Nagoya/Sapporo/Hachioji, 1996), Pitman Res. Notes Math. Ser., 381, 43-61. | MR | Zbl

[22] C. P. Rourke, - B. J. Sanderson, “Introduction to piecewise linear topology”, Springer Study edition, 1982. | MR | Zbl

[23] J. H. M. Steenbrink - J. Stevens, Topological invariance of the weight filtration, Indagationes Math. 46 (1984). | MR | Zbl

[24] J. H. M. Steenbrink, Mixed Hodge structures associated with isolated singularities, Proc. of Symp. in Pure Math. 40, Part 2 (1983), 513-536. | MR | Zbl