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

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 $(Y,\varnothing )$, $(X,Y)$, $(X,X-Y)$, $(X-Y,\varnothing )$ and $(\partial U,\varnothing )$. 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 $\partial U$.