Algebraic Intersection Spaces

TL;DR

This paper introduces algebraic intersection spaces applicable to a broad class of complex and real analytic spaces, extending previous theories and establishing duality and signature properties for these spaces.

Contribution

It extends intersection space theory to include many complex and real analytic spaces, proving their existence and duality properties, and defining a signature related to Novikov's invariant.

Findings

Algebraic intersection spaces exist for a wide class of spaces.

They satisfy duality when local obstructions vanish.

The signature matches Novikov's invariant.

Abstract

We define a variant of intersection space theory that applies to many compact complex and real analytic spaces $X$, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze "local duality obstructions", which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in $X$ of a tubular neighborhood of the singular set.