Intersection Space Constructible Complexes

Abstract

We present an obstruction theoretic inductive construction of intersection space pairs, which generalizes Banagl's construction of intersection spaces for arbitrary depth stratifications. We construct intersection space pairs for pseudomanifolds with compatible trivial structures at the link fibrations; this includes the case of toric varieties. We define intersection space complexes in an axiomatic way, similar to Goresky-McPherson axioms for intersection cohomology. We prove that if the intersection space exists, then the pseudomanifold has an intersection space complex whose hypercohomology recovers the cohomology of the intersection space pair. We characterize existence and uniqueness of intersection space complexes in terms of the derived category of constructible complexes. We show that intersection space complexes of algebraic varieties lift to the derived category of Mixed Hodge Modules, endowing intersection space cohomology with a Mixed Hodge Structure. We find classes of examples admitting intersection space complex, and counterexamples not admitting them; they are in particular the first examples known not admitting Banagl intersection spaces. We prove that the (shifted) Verdier dual of an intersection space complex is an intersection space complex. We prove a generic Poincare duality theorem for intersection space complexes.