Edges, Orbifolds, and Seiberg-Witten Theory

TL;DR

This paper applies Seiberg-Witten theory to derive new obstructions to Einstein metrics on 4-manifolds with specific conical singularities along surfaces, focusing on cone angles of the form 2(pi)/p.

Contribution

It introduces novel obstructions for Einstein metrics on 4-manifolds with orbifold singularities, expanding the understanding of geometric structures with conical singularities.

Findings

Obstructions are established for Einstein metrics with cone angles 2(pi)/p.

Results are specific to orbifold singularities along embedded surfaces.

Conjecture extends results to more general cone angles.

Abstract

Seiberg-Witten theory is used to obtain new obstructions to the existence of Einstein metrics on 4-manifolds with conical singularities along an embedded surface. In the present article, the cone angle is required to be of the form 2(pi)/p, where p is a positive integer, but we conjecture that similar results will also hold in greater generality.