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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 2π/p, p a positive integer, but we conjecture that similar results will also hold in greater generality. Recent work on Kähler–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 surface. These results are intimately tied to known phenomena in Kähler geometry, and reinforce the overarching principle that Kähler 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 Σ ⊂ M be a smoothly embedded compact surface. Near any point p ∈ Σ, we can thus find local coordinates (x, x, y, y) in which Σ is given by y = y = 0. Given any such adapted coordinate system, we then introduce an associated transversal polar coordinate system (ρ, θ, x, x) by setting y = ρ cos θ and y = ρ sin θ. Now fix some positive constant β > 0. An edge-cone metric g of cone angle 2πβ on (M, Σ) is a smooth Riemannian metric on M − Σ which takes the form g = dρ + βρ(dθ + ujdx) + wjkdxdx + ρh (1) in a suitable transversal polar coordinate system near each point of Σ, where the symmetric tensor field h on M is required to have infinite conormal regularity along Σ. This last assumption means that the components of h in (x, x, y, y) coordinates have continuous derivatives of all orders with respect to collections of smooth vector fields (e.g. ρ ∂/∂ρ, ∂/∂θ, ∂/∂x, ∂/∂x) which have vanishing normal component along Σ. Thus, an edge-cone metric g behaves like a smooth metric in directions parallel to Σ, but is modelled on a 2-dimensional cone in the transverse directions. An edge-cone metric g is said to be Einstein if its Ricci tensor r satisfies 2010 Mathematics Subject Classification. Primary 53C25; Secondary 53C21, 57R18, 57R57.