Link Bundles and Intersection Spaces of Complex Toric Varieties

TL;DR

This paper computes the rational homology of intersection spaces in complex 3D toric varieties, revealing that unlike intersection homology, it encodes more refined fan data and is not purely combinatorial.

Contribution

It introduces a method to compute intersection space homology for 3D toric varieties and compares it to intersection homology, highlighting differences in invariance properties.

Findings

Intersection space homology is not combinatorially invariant.

Homology of links in 3D toric varieties is explicitly computed.

Intersection space homology captures more refined topological information.

Abstract

There exist several homology theories for singular spaces that satisfy generalized Poincar\'e duality, including Goresky-MacPherson's intersection homology, Cheeger's $L^2$ cohomology and the homology of intersection spaces. The intersection homology and $L^2$ cohomology of toric varieties is known. Here, we compute the rational homology of intersection spaces of complex 3-dimensional toric varieties and compare it to intersection homology. To achieve this, we analyze cell structures and topological stratifications of these varieties and determine compatible structures on their singularity links. In particular, we compute the homology of links in 3-dimensional toric varieties. We find it convenient to use the concept of a rational homology stratification. It turns out that the intersection space homology of a toric variety, contrary to its intersection homology, is not combinatorially invariant and thus retains more refined information on the defining fan.