Lowest Weights in Cohomology of Variations of Hodge Structure

TL;DR

This paper demonstrates that the image of a natural map between intersection cohomology and cohomology of variations of Hodge structure captures the lowest weight component of the mixed Hodge structure, using Saito's theory.

Contribution

It establishes a precise relationship between the image of a natural map and the lowest weight part of the mixed Hodge structure in the context of variations of Hodge structure.

Findings

The image of the map equals the lowest weight part of the mixed Hodge structure.

Uses Saito's theory of mixed Hodge modules for the proof.

Clarifies the weight filtration in the cohomology of variations of Hodge structure.

Abstract

Let X be a smooth complex projective variety, let $j:U\into X$ an immersion of a Zariski open subset, and let V be a variation of Hodge structure of weight n over U. Then IH^k(X, j_*V) is known to carry a pure Hodge structure of weight k+n, while H^k(U,V) carries a mixed Hodge structure of weight $\ge k+n$. In this note it is shown that the image of the natural map $IH^k(X,j_*V) \to H^k(U,V)$ is the lowest weight part of this mixed Hodge structure. The proof uses Saito's theory of mixed Hodge modules.