  We study the problem of vortex solutions in the background of rotating black
holes in both asymptotically flat and asymptoticlly anti de Sitter spacetimes.
We demonstrate the Abelian Higgs field equations in the background of four
dimensional Kerr, Kerr-AdS and Reissner-Nordstrom-AdS black holes have vortex
line solutions. These solutions, which have axial symmetry, are generalization
of the Nielsen-Olesen string. By numerically solving the field equations in
each case, we find that these black holes can support an Abelian Higgs field as
hair. This situation holds even in the extremal case, and no flux-expulsion
occurs. We also compute the effect of the self gravity of the Abelian Higgs
field show that the the vortex induces a deficit angle in the corresponding
black hole metrics.
