  Appreciable star formation, and, therefore, numerous massive stars, are
frequently found near supermassive black holes (SMBHs). As a result,
core-collapse supernovae in these regions should also be expected. In this
paper, we consider the observational consequences of predicting the fate of
supernova remnants (SNRs) in the sphere of influence of quiescent SMBHs. We
present these results in the context of `autarkic' nuclei, a model that
describes quiescent nuclei as steady-state and self-sufficient environments
where the SMBH accretes stellar winds with no appreciable inflow of material
from beyond the sphere of influence. These regions have properties such as gas
density that scale with the mass of the SMBH. Using predictions of the X-ray
lifetimes of SNRs originating in the sphere of influence, we make estimates of
the number of core collapse SNRs present at a given time. With the knowledge of
lifetimes of SNRs and their association with young stars, we predict a number
of core-collapse SNRs that grows from ~1 around Milky Way-like (4.3 x 10^6
Msun) SMBHs to ~100 around the highest mass (10^10 Msun) SMBHs. The presence of
young SNRs will amplify the X-ray emission near quiescent SMBHs, and we show
that the total core-collapse SNR emission has the potential to influence soft
X-ray searches for very low-luminosity SMBHs. Our SNR lifetime estimates also
allow us to predict star formation rates in these regions. Assuming a
steady-state replenishment of massive stars, we estimate a star formation rate
density of 2 x 10^-4 Msun/yr/pc^2 around the Milky Way SMBH, and a similar
value around other SMBHs due to a weak dependence on SMBH mass. This value is
consistent with currently available observations.
