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 \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(Z) is pure. The main ingredient of the proof is purity of the intersection cohomology sheaf.