Some differential complexes within and beyond parabolic geometry

TL;DR

This paper introduces new differential complexes for manifolds with geometric structures, generalizing the Bernstein-Gelfand-Gelfand complex for parabolic geometries and extending to some non-parabolic cases.

Contribution

It provides simpler constructions of complexes that serve as fine resolutions of locally constant functions, avoiding complex machinery and extending beyond parabolic geometries.

Findings

Constructed complexes that replace the de Rham complex.

Complexes coincide with Bernstein-Gelfand-Gelfand complexes in parabolic cases.

Method extends to certain non-parabolic geometries.

Abstract

For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein-Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to certain geometries beyond the parabolic realm.