  The generation and evolution of magnetic fields in the plasma accreting into
a rotating black hole is studied in the 3+1 split of the Kerr metric. Attention
is focused on effects of the gravitomagnetic potential. The gravitomagnetic
force appears as battery term in the generalized Ohm's law. The gravitomagnetic
battery is likely to saturate at higher field strength than the classical
Biermann battery.
  The coupling of the gravitomagnetic potential with electric fields appears as
gravitomagnetic current in Maxwell's equations. In the magnetohydrodynamic
induction equation, this current re-appears as source term for the poloidal
magnetic field, which can produce closed magnetic structures around an
accreting black hole. In principle, even self-excited axisymmetric dynamo
action is possible, which means that Cowling's anti-dynamo theorem does not
hold in the Kerr metric.
  Finally, simulations of the alpha-Omega dynamo in accretion flows into the
hole are presented. I assume a simple expression of alpha in this relativistic
context. The modes of the dynamo are oscillating for dynamo numbers which are
typical for accretion disks. In a zero angular momentum flow into a Kerr black
hole there is still shear, i.e. the angular velocity, Omega, of the plasma
equals the angular velocity of space, omega, and it has been speculated (Meier
1998) that even then a dynamo could operate. This is shown to be unlikely due
to the rapid accretion.
