Weight filtration on the cohomology of algebraic varieties

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 $X^{\prime }$ 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 $X^{\prime }$. 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.

arXiv comment11 pages, title is changed

Cited by in corpus (1)

• Boundary complexes and weight filtrations