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 × 106 M⊙) SMBHs to ∼100 around the highest mass (1010 M⊙) 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 × 10-4 M⊙ yr-1 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.
