  We examined analytically and proposed a numerical model of an Abelian Higgs
dark matter vortex in the spacetime of a stationary axisymmetric Kerr black
hole. In analytical calculations the dark matter sector was modeled by an
addition of a U (1)-gauge field coupled to the visible sector. The backreaction
analysis revealed that the impact of the dark vortex presence is far more
complicated than causing only a deficit angle. The vortex causes an ergosphere
shift and the event horizon velocity is also influenced by its presence. These
phenomena are more significant than in the case of a visible vortex sector. The
area of the event horizon of a black hole is diminished and this decline is
larger in comparison to the Kerr black hole with an Abelian Higgs vortex case.
After analyzing the gravitational properties for the general setup, we focused
on the subset of models that are motivated by particle physics. We retained the
Abelian Higgs model as a description of the dark matter sector (this sector
contained a heavy dark photon and an additional complex scalar) and added a
real scalar representing the real component of the Higgs doublet in the unitary
gauge, as well as an additional U (1)-gauge field representing an ordinary
electromagnetic field. Moreover, we considered two coupling channels between
the visible and dark sectors, which were the kinetic mixing between the gauge
fields and a quartic coupling between the scalar fields. After solving the
equations of motion for the matter fields numerically we analyzed properties of
the cosmic string in the dark matter sector and its influence on the visible
sector fields that are directly coupled to it. We found out that the presence
of the cosmic string induced spatial variation in the vacuum expectation value
of the Higgs field and a nonzero electromagnetic field around the black hole.
