On the weight filtration of the homology of algebraic varieties : the generalized Leray cycles

Elzein, Fouad; Némethi, András

Elzein, Fouad; Némethi, András

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 4, p. 869-903

Abstract

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 $Y \subset X$ , 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 $(Y, \emptyset)$ , $(X, Y)$ , $(X, X - Y)$ , $(X - Y, \emptyset)$ and $(\partial U, \emptyset)$ . The construction is compatible with the weights in $H_{*} (A, B, ℚ)$ 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 $\partial U$ .

MR 1991006 | Zbl 1098.14006

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},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {4},
     year = {2002},
     pages = {869-903},
     zbl = {1098.14006},
     mrnumber = {1991006},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/ASNSP_2002_5_1_4_869_0/

References

[1] N. A'Campo, La fonction zéta d'une monodromie, Commentarii Mathematici Helvetici 50 (1975), 233-248. | MR 371889 | Zbl 0333.14008

[2] A. Borel et al., “Seminar on Intersection Cohomology”, Forschungsinstitut für Mathematik ETH Zürich, 1984. | Zbl 0553.14002

[3] G. E. Bredon, Topology and Geometry, In: “Graduate Texts in Mathematics”, 139, Springer 1993. | MR 1224675 | Zbl 0791.55001

[4] G. E. Bredon, Sheaf Theory, In: “Graduate Texts in Mathematics”, 170, Springer 1997. | MR 1481706 | Zbl 0874.55001

[5] C. H. Clemens, Degeneration of Kähler manifolds, Duke Math. Journal 44 (1977), 215-290. | MR 444662 | Zbl 0353.14005

[6] P. Deligne, Théorie de Hodge, II, III, Publ. Math. IHES 40, 44 (1972, 1975), 5-47, 6-77. | Numdam | MR 498551 | Zbl 0237.14003

[7] F. El Zein, Introduction à la théorie de Hodge mixed, In: “Actualiés Mathématiques”, Hermann, Paris 1991. | MR 1160988 | Zbl 0718.58001

[8] A. Fujiki, Duality of Mixed Hodge Structures of Algebraic Varieties, Publ. RIMS, Kyoto Univ., 16 (1980), 635-667. | MR 602463 | Zbl 0475.14006

[9] M. Goresky - R. Macpherson, Intersection Homology Theory, Topology, 19 (1980), 135-162. | MR 572580 | Zbl 0448.55004

[10] M. Goresky - R. Macpherson, Intersection Homology II, Invent. Math. 71 (1983), 77-129. | MR 696691 | Zbl 0529.55007

[11] P. Griffiths - W. Schmid, Recent developments in Hodge Theory, In: “Discrete subgroups of Lie groups”, Bombay Colloquium, Oxford Univ. Press, 1973. | Zbl 0355.14003

[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 0638.00011

[13] N. Habegger - L. Saper, Intersection cohomology of cs-spaces and Zeeman's filtration, Invent. Math. 105 (1991), 247-272. | MR 1115543 | Zbl 0759.55003

[14] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79 (1964), 109-326. | MR 199184 | Zbl 0122.38603

[15] H. Hironaka, Triangulations of algebraic sets, Proceedings of Symposia in Pure Mathematics 29 (1975), 165-185. | MR 374131 | Zbl 0332.14001

[16] H. C. King, Topological invariance of intersection homology without sheaves, Topology and its applications 20 (1985), 149-160. | MR 800845 | Zbl 0568.55003

[17] J. Leray, Problème de Cauchy III, Bull. Soc. Math. France 87 (1959), 81-180. | Numdam | MR 125984 | Zbl 0199.41203

[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 713250 | Zbl 0538.14014

[20] V. Navarro Aznar, Sur la Théorie de Hodge des Variétés Algébriques à Singularités Isolées, Astérisque 130 (1985), 272-305. | MR 804059 | Zbl 0599.14007

[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 1607674 | Zbl 0899.32014

[22] C. P. Rourke, - B. J. Sanderson, “Introduction to piecewise linear topology”, Springer Study edition, 1982. | MR 665919 | Zbl 0477.57003

[23] J. H. M. Steenbrink - J. Stevens, Topological invariance of the weight filtration, Indagationes Math. 46 (1984). | MR 748980 | Zbl 0539.14016

[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 713277 | Zbl 0515.14003