  (Abridged) We complete the census of nuclear X-ray activity in 100 early type
Virgo galaxies observed by the Chandra X-ray Telescope as part of the
AMUSE-Virgo survey, down to a (3sigma) limiting luminosity of 3.7E+38 erg/s
over 0.5-7 keV. The stellar mass distribution of the targeted sample, which is
mostly composed of formally `inactive' galaxies, peaks below 1E+10 M_Sun, a
regime where the very existence of nuclear super-massive black holes (SMBHs) is
debated. Out of 100 objects, 32 show a nuclear X-ray source, including 6 hybrid
nuclei which also host a massive nuclear cluster as visible from archival HST
images. After carefully accounting for contamination from nuclear low-mass
X-ray binaries based on the shape and normalization of their X-ray luminosity
function, we conclude that between 24-34% of the galaxies in our sample host a
X-ray active SMBH (at the 95% C.L.). This sets a firm lower limit to the black
hole occupation fraction in nearby bulges within a cluster environment. At face
value, the active fraction -down to our luminosity limit- is found to increase
with host stellar mass. However, taking into account selection effects, we find
that the average Eddington-scaled X-ray luminosity scales with black hole mass
as M_BH^(-0.62^{+0.13}_{-0.12}), with an intrinsic scatter of
0.46^({+0.08}_{-0.06}) dex. This finding can be interpreted as observational
evidence for `down-sizing' of black hole accretion in local early types, that
is, low mass black holes shine relatively closer to their Eddington limit than
higher mass objects. As a consequence, the fraction of active galaxies, defined
as those above a fixed X-ray Eddington ratio, decreases with increasing black
hole mass.
