Branchfolds and rational conifolds

TL;DR

This paper introduces branchfolds, a generalization of orbifolds to include rational cone singularities, and establishes their connection with branched coverings and conifold structures.

Contribution

It extends orbifold theory to branchfolds, providing a geometric characterization and linking conifold structures with locally finite holonomy.

Findings

Branchfolds generalize orbifolds to rational cone singularities.

A conifold admits a branchfold structure iff it has locally finite holonomy.

The paper proves a geometric goodness theorem for branchfolds.

Abstract

We extend the concept of orbifold to that of branchfold, in order to allow any cone singularities with rational angles, and show why branchfolds naturally fit in the theory of branched coverings. Then, we obtain a geometric goodness theorem for branchfolds and apply it to prove that a conifold can be endowed with branchfold structure if and only if it has locally finite holonomy.