Weight filtration on the cohomology of algebraic varieties

TL;DR

This paper establishes the existence of a canonical weight filtration on the etale cohomology of algebraic varieties and relates the middle weight part to intersection cohomology, extending known results to more general cases.

Contribution

It proves the weight filtration exists for any algebraic variety and connects the middle weight part to intersection cohomology using Gabber's purity theorem.

Findings

The weight filtration W exists on etale cohomology with compact supports.

The middle weight part is a subspace of intersection cohomology.

The result generalizes Weber's case for proper varieties over complex numbers.

Abstract

We show that the etale cohomology (with compact supports) of an algebraic variety $X$ over an algebraically closed field has the canonical weight filtration $W$, and prove that the middle weight part of the cohomology with compact supports of $X$ is a subspace of the intersection cohomology of a compactification $X'$ of X, or equivalently, the middle weight part of the (so-called) Borel-Moore homology of $X$ is a quotient of the intersection cohomology of $X'$. We are informed that this has been shown by A. Weber in the case $X$ is proper (and $k=\bC$) using a theorem of G. Barthel, J.-P. Brasselet, K.-H. Fieseler, O. Gabber and L. Kaup on morphisms between intersection complexes. We show that the assertion immediately follows from Gabber's purity theorem for intersection complexes.