  We investigate the dynamics and emission of a starved magnetospheric region
(gap) formed in the vicinity of a Kerr black hole horizon, using a new, fully
general relativistic particle-in-cell code that implements Monte Carlo methods
to compute gamma-ray emission and pair production through the interaction of
pairs and gamma rays with soft photons emitted by the accretion flow. It is
found that when the Thomson length for collision with disk photons exceeds the
gap width, screening of the gap occurs through low-amplitude, rapid plasma
oscillations that produce self-sustained pair cascades, with quasi-stationary
pair and gamma-ray spectra, and with a pair multiplicity that increases in
proportion to the pair production opacity. The gamma-ray spectrum emitted from
the gap peaks in the TeV band, with a total luminosity that constitutes a
fraction of about $10^{-5}$ of the corresponding Blandford-Znajek power. This
stage is preceded by a prompt discharge phase of duration $\sim r_g/c$, during
which the potential energy initially stored in the gap is released as a flare
of curvature TeV photons. We speculate that the TeV emission observed in M87
may be produced by pair discharges in a spark gap.
