  We present a new class of near-horizon geometries which solve Einstein's
vacuum equations, including a negative cosmological constant, in all even
dimensions greater than four. Spatial sections of the horizon are inhomogeneous
S^2-bundles over any compact Kaehler-Einstein manifold. For a given base, the
solutions are parameterised by one continuous parameter (the angular momentum)
and an integer which determines the topology of the horizon. In six dimensions
the horizon topology is either S^2 x S^2 or CP^2 # -CP^2. In higher dimensions
the S^2-bundles are always non-trivial, and for a fixed base, give an infinite
number of distinct horizon topologies. Furthermore, depending on the choice of
base we can get examples of near-horizon geometries with a single rotational
symmetry (the minimal dimension for this is eight). All of our horizon
geometries are consistent with all known topology and symmetry constraints for
the horizons of asymptotically flat or globally Anti de Sitter extremal black
holes.
