Self-dual intersection space complexes

TL;DR

This paper constructs a canonical self-dual intersection space sheaf complex for Witt spaces with trivialized link bundles, establishing a new intersection cohomology theory satisfying Poincaré duality for complex stratified spaces.

Contribution

It introduces K"unneth complexes and proves the existence and uniqueness of self-dual intersection space complexes, extending intersection homology concepts to new classes of pseudomanifolds.

Findings

Existence of a canonical Verdier self-dual intersection space sheaf complex.

Development of a new intersection cohomology theory satisfying Poincaré duality.

Introduction of K"unneth complexes with triviality structures along strata.

Abstract

In this article, we prove that there is a canonical Verdier self-dual intersection space sheaf complex for the middle perversity on Witt spaces that admit compatible trivializations for their link bundles, for example toric varieties. If the space is an algebraic variety our construction takes place in the category of mixed Hodge modules. We obtain an intersection space cohomology theory, satisfying Poincar\'e duality, valid for a class of pseudomanifolds with arbitrary depth stratifications. The main new ingredient is the category of K\"unneth complexes; these are cohomologically constructible complexes with respect to a fixed stratification, together with additional data, which codifies triviality structures along the strata. In analogy to what Goreski and McPherson showed for intersection homology complexes, we prove that there are unique K\"unneth complexes that satisfy the axioms for intersection space complexes introduced by the first and third author. This uniqueness implies the duality statements in the same scheme as in Goreski and McPherson theory.