Lowest weights in cohomology of variations of Hodge structure

Abstract

Let $X$ be an irreducible complex analytic space with $j:U\hookrightarrow X$ an immersion of a smooth Zariski-open subset, and let $\mathbb{V}$ be a variation of Hodge structure of weight $n$ over $U$. Assume that $X$ is compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent, $I{H}^k(X,\mathbb{V})$ is known to carry a pure Hodge structure of weight $k+n$, while $H^k(U,\mathbb{V})$ carries a mixed Hodge structure of weight at least $k+n$. In this note it is shown that the image of the natural map $I{H}^k(X,\mathbb{V})\to H^k(U,\mathbb{V})$ is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement $X-U$ is not a hypersurface.