Intersection Space Constructible Complexes

TL;DR

This paper develops a generalized, axiomatic framework for constructing intersection space complexes in stratified pseudomanifolds, extending Banagl's work and connecting to mixed Hodge structures.

Contribution

It introduces an axiomatic approach to intersection space complexes, characterizes their existence and uniqueness, and explores their properties and applications in algebraic geometry.

Findings

Constructs intersection space pairs for pseudomanifolds with trivial link fibrations.

Defines intersection space complexes and proves their hypercohomology recovers intersection space cohomology.

Identifies classes of varieties with and without intersection space complexes, including counterexamples.

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.