[math-fun] Black torus?

Can a fast rotating "black hole" rotate fast enough to be a "black torus" ? I'll accept an unstable equilibrium black torus for the moment; what I'm asking is whether the event horizon can be toroidal for some finite amount of time. The real issue is whether physicist Eve could theoretically go through the center of such a torus without getting gobbled up by the black object.

From an infinitesimal point of view it seems possible; after all, sufficiently small neighborhoods on the toroidal surface look like small neighborhoods on the surface of a black hole.