Purity of boundaries of open complex varieties

Abstract

We study the boundary of an open smooth complex algebraic variety $U$. We ask when the cohomology of the geometric boundary $Z=X\setminus U$ in a smooth compactification $X$ is pure with respect to the mixed Hodge structure. Knowing the dimension of singularity locus of some singular compactification we give a bound for $k$ above which the cohomology $H^k(Z)$ is pure. The main ingredient of the proof is purity of the intersection cohomology sheaf.