  M87 is a nearby radio galaxy that has been detected at energies ranging from
radio to very high energy (VHE) gamma-rays. Its proximity and its jet,
misaligned from the line of sight allow detailed morphological studies. The
imaging atmospheric Cherenkov technique (from 100 GeV to 10 TeV) provides
insufficient angular resolution (few arc-minutes) to resolve the M87 emission
region. However, the short time scale variability observed by MAGIC, HESS and
VERITAS suggests the TeV emission is coming from a very small region, most
likely close to the core. We propose that the variable TeV emission may be
produced in a pair-starved region of the central black hole (BH) magnetosphere,
i.e. a region where the density of the electron-positron plasma is not
sufficient to completely screen the accelerating electric field. The funnel, a
low density and magnetically dominated region around the poles, appears as a
favourable site of low-density where a Blandford-Znajek process may explain the
main properties of the TeV $\gamma$-ray emission from M87. We produce a
broadband spectral energy distribution (SED) of the resulting radiation and
compare the model with the observed fluxes from the nucleus of M87, for both
low and high $\gamma$-ray activities. We finish with a brief discussion on the
connection between the accretion rate and the intermittence in the formation of
gaps in the magnetosphere.
