Lowest Weights in 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.