Self-dual intersection space complexes

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 Poincaré duality, valid for a class of pseudomanifolds with arbitrary depth stratifications. The main new ingredient is the category of Künneth complexes; these are cohomologically constructuble 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 Künneth 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.

A generalized de Rham Theorem for Intersection Spaces

“…Let X = M ∪ ∂M (Σ × cone(L)) and let T = (Σ × cone(L)) denote the tube of Σ ⊂ X. We denote by t : T \ Σ = Σ × L × (0, 1) → X \ Σ the smooth embedding of the tube into the regular part of X and by j : Σ × L → T the embedding at 1 2 . Then, for any perversity function p, the ΩI • p -complex is defined as follows.…”

“…In ongoing work, the de Rham model for greater stratification depth, which was introduced by the author in [9], is compared to the cohomology of Agustin and Bobadilla's intersection spaces. The results of [1] are important tools in this setting, especially the uniqueness results when passing to the derived category.…”