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

Fouad Elzein; András Némethi

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 4, page 869-903
  • ISSN: 0391-173X

Abstract

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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 ⊂ 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 , ∅ ) , ( X , Y ) , ( X , X - Y ) , ( X - Y , ∅ ) and ( ∂ U , ∅ ) . 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 ∂ U .

How to cite

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Elzein, Fouad, and 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 1.4 (2002): 869-903. <http://eudml.org/doc/84490>.

@article{Elzein2002,
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 $H_*(A,B,\{\mathbb \{Q\}\})$ 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 $\partial U$.},
author = {Elzein, Fouad, Némethi, András},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {869-903},
publisher = {Scuola normale superiore},
title = {On the weight filtration of the homology of algebraic varieties : the generalized Leray cycles},
url = {http://eudml.org/doc/84490},
volume = {1},
year = {2002},
}

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
PB - Scuola normale superiore
VL - 1
IS - 4
SP - 869
EP - 903
AB - 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 $H_*(A,B,{\mathbb {Q}})$ 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 $\partial U$.
LA - eng
UR - http://eudml.org/doc/84490
ER -

References

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