  We study Einstein metrics on smooth compact 4-manifolds with an edge-cone
singularity of specified cone angle along an embedded 2-manifold. To do so, we
first derive modified versions of the Gauss-Bonnet and signature theorems for
arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show
that these yield non-trivial obstructions in the Einstein case. We then use
these integral formulae to obtain interesting information regarding
gravitational instantons which arise as limits of such edge-cone manifolds.
