An exploration of connections and curvature in the presence of singularities

TL;DR

This paper extends the concepts of connections and curvature to Lie-Rinehart algebras without smoothness, exploring their existence, gauge transformations, and applications to singular varieties with non-zero curvature.

Contribution

It introduces a framework for defining connections and curvature in non-smooth settings, including singular varieties, and relates these to gauge transformations and Chern characters.

Findings

Connections can exist on finitely generated modules over singular spaces.

Gauge transformations act on the space of connections.

Levi-Civita connections with non-zero curvature are constructed on singular varieties.

Abstract

We develop the notions of connections and curvature for general Lie-Rinehart algebras without using smoothness assumptions on the base space. We present situations when a connection exists. E.g., this is the case when the underlying module is finitely generated. We show how the group of module automorphism acts as gauge transformations on the space of connections. When the underlying module is projective we define a version of the Chern character reproducing results of Hideki Ozeki. We discuss various examples of flat connections and the associated Maurer-Cartan equations. We provide examples of Levi-Civita connections on singular varieties and singular differential spaces with non-zero Riemannian curvature. The main observation is that for quotient singularities, even though the metric degenerates along strata, the poles of the Christoffel symbols are removable.