Topology of algebraic varieties

Let Y be a normal crossing divisor in the smooth projective algebraic variety X (defined over ${\mathbb C}$) and let U be a tubular neighbourhood of Y in X. We construct homological cycles generating $H_*(A,B)$, 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.