Topology of algebraic varieties

Fouad Elzein and András Némethi

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.