Lowest Weights in Cohomology of Variations of Hodge Structure (II)

TL;DR

This paper proves that in the setting of complex analytic spaces with variations of Hodge structure, the image of the intersection cohomology in the usual cohomology corresponds exactly to the lowest weight part of the mixed Hodge structure, extending known algebraic results to the analytic case.

Contribution

It establishes that the image of the natural map from intersection cohomology to cohomology captures the lowest weight component of the mixed Hodge structure in the analytic setting.

Findings

The image of IH^k(X, V) in H^k(U, V) is the lowest weight part of the mixed Hodge structure.

This result extends algebraic case conclusions to complex analytic spaces, even when the complement is not a hypersurface.

The proof involves handling the complexities of mixed sheaves in the analytic context.

Abstract

Let $X$ be an irreducible complex analytic space with $j:U\into X$ an immersion of a smooth Zariski open subset, and let $\bV$ be a variation of Hodge structure of weight $n$ over $U$. Assume $X$ is compact K\"ahler. Then provided the local monodromy operators at infinity are quasi-unipotent, $IH^k(X, \bV)$ is known to carry a pure Hodge structure of weight $k+n$, while $H^k(U,\bV)$ carries a mixed Hodge structure of weight $\ge k+n$. In this note it is shown that the image of the natural map $IH^k(X,\bV) \to H^k(U,\bV)$ 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.