  Context: Accreting black holes (BHs) may be surrounded by a highly magnetized
plasma threaded by a poloidal magnetic field. Non-thermal flares and high
energy components could originate from a hot, collisionless and nearly
force-free corona. The jets we often observe from these systems are believed to
be rotation-powered and magnetically-driven.
  Aims: We study axisymmetric BH magnetospheres where some magnetic field lines
anchored in a surrounding disk can connect to the event horizon of a rotating
BH. We identify the sites of magnetic reconnection within 30 gravitational
radii depending on the BH spin.
  Methods: With the fully general relativistic particle-in-cell code GRZeltron,
we solve the time-dependent dynamics of the electron-positron pair plasma and
of the electromagnetic fields around the BH. The disk is represented by a
steady plasma in Keplerian rotation, threaded by a frozen dipolar field.
  Results: For prograde disks, twisted open magnetic field lines crossing the
horizon power a Blandford-Znajek jet while beyond a critical distance, open
field lines on the disk are open. In the innermost regions, coupling field
lines ensure the transfer of significant amounts of angular momentum and energy
between the BH and the disk. From the Y-point at the intersection, a current
sheet forms where particle acceleration via magnetic reconnection takes place.
We compute the synchrotron images of the current sheet emission.
  Conclusions: Our estimates for jet power and BH-disk exchanges match those
derived from purely force-free models. Dissipation at the Y-point heats the
corona and provides a physically motivated source of hard X-rays above the disk
for reflection models. Episodic plasmoid ejection might explain millisecond
flares observed in Cyg X-1. Particles flowing from the Y-point down to the disk
could produce a hot spot at the footpoint of the outermost closed field line.
