Purity at the end

TL;DR

This paper investigates the cohomological purity of the boundary at infinity for smooth algebraic manifolds, utilizing perverse sheaves to analyze the topological properties of their completions.

Contribution

It introduces new methods to prove cohomological purity of the boundary at infinity using perverse sheaves and studies related topological properties.

Findings

Proved purity of cohomology at infinity for smooth algebraic manifolds.

Established connections between singular completions and cohomological properties.

Provided new insights into the topology of algebraic manifold completions.

Abstract

We consider smooth completion of algebraic manifolds. Having some information about its singular completions or about completions of its images we prove purity of cohohomology of the set at infinity. We deduce also some topological properties. The work is based on the study of perverse direct images for algebraic maps.