The standard filtration on cohomology with compact supports with an appendix on the base change map and the Lefschetz hyperplane theorem

TL;DR

This paper explores the structure of cohomology with compact supports on quasi-projective varieties, utilizing filtrations, hyperplane sections, and perverse sheaves, with implications for the Lefschetz hyperplane theorem and base change maps.

Contribution

It provides a detailed description of the standard and Leray filtrations on cohomology with compact supports, employing hyperplane sections and perverse sheaves, and discusses base change maps in this context.

Findings

Describes the standard and Leray filtrations on cohomology with compact supports.

Uses hyperplane sections to analyze filtrations on quasi-projective varieties.

Connects base change maps with the Lefschetz hyperplane theorem for constructible sheaves.

Abstract

We describe the standard and Leray filtrations on the cohomology groups with compact supports of a quasi projective variety with coefficients in a constructible complex using flags of hyperplane sections on a partial compactification of a related variety. One of the key ingredients of the proof is the Lefschetz hyperplane theorem for perverse sheaves and, in an appendix, we discuss the base change maps for constructible sheaves on algebraic varieties and their role in a proof, due to Beilinson, of the Lefschetz hyperplane theorem.