# A Topological Characterization of the Middle Perversity Intersection Complex for Arbitrary Complex Algebraic Varieties

## TL;DR

This paper provides topological characterizations of the middle perversity intersection complex for complex algebraic varieties, establishing its invariance under homeomorphisms and extending Deligne's construction to stratified spaces.

## Contribution

It introduces two axiomatic topological characterizations of the intersection complex, applicable to arbitrary complex algebraic varieties, and demonstrates its invariance under homeomorphisms.

## Key findings

- Topological axiomatic characterizations of the intersection complex.
- Invariance of the intersection complex under homeomorphisms.
- Extension of Deligne's construction to stratified spaces.

## Abstract

For an arbitrary complex algebraic variety which is not necessarily pure dimensional, the intersection complex can be defined as the direct sum of the Deligne-Goresky-MacPherson intersection complexes of each irreducible component. We give two axiomatic topological characterizations of the middle perversity direct sum intersection complex, one stratification dependent and the other stratification independent. To accomplish this, we show that this direct sum intersection complex can be constructed using Deligne's construction in the more general context of topologically stratified spaces. A consequence of these characterizations is the invariance of this direct sum intersection complex under homeomorphisms.