Topology of algebraic varieties

Abstract

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 $H_{\ast }(A,B)$, 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 $H_{\ast }(A,B,\mathbb{Q})$ of Deligne's mixed Hodge structure.