  The replica trick defines Renyi entropies as partition functions on conically
singular geometries. We discuss their gravity duals: regular bulk solutions to
the Einstein equations inducing conically singular metrics at the boundary.
When the conical singularity is supported on a flat or spherical surface, these
solutions are rewritings of the hyperbolic black hole. For more general shapes,
these solutions are new. We construct them perturbatively in a double expansion
in the distance and strength of the conical singularity, and extract the vacuum
polarisation due to the cone. Recent results about the structure of logarithmic
divergences of Renyi entropies are reproduced ---in particular, $f_b\neq f_c$.
We discuss in detail the dynamical resolution of the singularity in the bulk.
This resolution is in agreement with a previous proposal, and indicates a
non-minimal settling to the `splitting problem': an apparent ambiguity in the
holographic entropy formula of certain theories with higher derivatives.
