Purity of boundaries of open complex varieties

TL;DR

This paper investigates the conditions under which the cohomology of the boundary of an open complex algebraic variety is pure, providing bounds based on singularity dimensions and utilizing intersection cohomology sheaf purity.

Contribution

It offers new bounds for the purity of boundary cohomology in open complex varieties using intersection cohomology and singularity dimension analysis.

Findings

Bound for cohomology purity based on singularity dimension

Purity of intersection cohomology sheaf as a key tool

Criteria for when boundary cohomology is pure

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.