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, Série 5, Tome 1 (2002) no. 4, pp. 869-903.

Résumé

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 $\partial U$ .

EuDML MR Zbl

Classification : 14C30, 14F25

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     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},
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     zbl = {1098.14006},
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     url = {http://www.numdam.org/item/ASNSP_2002_5_1_4_869_0/}
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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, Série 5, Tome 1 (2002) no. 4, pp. 869-903. http://www.numdam.org/item/ASNSP_2002_5_1_4_869_0/

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