Edges, orbifolds, and Seiberg–Witten theory

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…=p, p a positive integer, but we conjecture that similar results will also hold in greater generality. Recent work on Kahler-Einstein metrics by Chen, Donaldson, Sun, and others (5), (12), (13), (17), (26) has elicited wide interest in the existence and uniqueness problems for Einstein metrics with conical singularities along a submanifold of real codimension 2. This article will show that Seiberg-Witten theory gives rise to interesting obstructions to the existence of 4-dimensional Einstein metrics with conical singularities along a sur- face. These results are intimately tied to known phenomena in Kahler geometry, and reinforce the overarching principle that Kahler metrics play a uniquely privileged role in 4-dimensional Riemannian geometry, to a degree that is simply unparalleled in other dimensions. We now recall the definition (2) of an edge-cone metric on a 4-manifold. Let M be a smooth compact 4-manifold, let § ‰ M be a smoothly embedded compact surface. Near any point p 2 §, we can thus find local coordinates (x 1 ;x 2 ;y 1 ;y 2 ) in which § is given by y 1 = y 2 = 0. Given any such adapted coordinate system, we then introduce an associated transversal polar coordinate system (‰;µ;x 1 ;x 2 ) by setting y 1 = ‰cosµ and y 2 = ‰sinµ. Now fix some positive constant fl > 0. An edge-cone metric g of cone angle