On the action of algebraic correspondences on weight spectral sequences

TL;DR

This paper develops an intersection theoretic method to analyze how algebraic correspondences act on weight spectral sequences, facilitating computations and compatibility proofs crucial for applications to Shimura varieties.

Contribution

It introduces a new intersection theoretic construction of correspondences on weight spectral sequences, avoiding blow-ups and proving their compatibility with compositions.

Findings

Provides a practical method for computing correspondences without blow-ups

Establishes compatibility of correspondences with compositions

Enables applications to Shimura varieties

Abstract

In a work of T. Saito, the action of algebraic correspondences on the etale cohomology of varieties over local fields with semistable reduction is related to correspondences on smaller strata via weight spectral sequences. We give an intersection theoretic construction of these correspondences. Under a finiteness condition this enables us to compute them without involving the blow-ups of products, and prove their compatibility with compositions. These features are essential for the application to Shimura varieties.