BGG resolutions via configuration spaces

TL;DR

This paper investigates the blow-ups of configuration spaces, introducing Orlik-Solomon manifolds to compute intersection cohomology and geometrically realize the sl_2 BGG resolution as an Aomoto complex.

Contribution

It provides a new geometric approach to the BGG resolution using configuration space blow-ups and Orlik-Solomon manifolds.

Findings

Computed intersection cohomology of flat connections with logarithmic singularities.

Realized the sl_2 BGG resolution as an Aomoto complex.

Established a geometric framework connecting configuration spaces and representation theory.

Abstract

We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik-Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the sl_2 Bernstein - Gelfand - Gelfand resolution as an Aomoto complex.