The MBHBM$^{\star}$ Project -- II. Molecular Gas Kinematics in the Lenticular Galaxy NGC 3593 Reveal a Supermassive Black Hole
As part of the Measuring Black Holes in Below Milky Way-mass (M$^\star$) galaxies (MBHBM$^\star$) Project, we present a dynamical measurement of the supermassive black hole (SMBH) mass in the nearby lenticular galaxy NGC 3593, using cold molecular gas $^{12}$CO(2-1) emission observed at an angular resolution of $\approx0''.3$ ($\approx10$ pc) with the Atacama Large Millimeter/submillimeter Array (ALMA). Our ALMA observations reveal a circumnuclear molecular gas disc (CND) elongated along the galaxy major axis and rotating around the SMBH. Using dynamical modelling, the molecular gas kinematics allow us to infer a SMBH mass $M_{\rm BH}=2.40_{-1.05}^{+1.87}\times10^6$ M$_\odot$ (only statistical uncertainties at the $3σ$ level). We also detect a massive core of cold molecular gas (CMC) of mass $M_{\rm CMC}=(5.4\pm1.2)\times10^6$ M$_\odot$ and effective (half-mass) radius $r_{\rm CMC,e}=11.2\pm2.8$ pc, co-spatial with a nuclear star cluster (NSC) of mass $M_{\rm NSC}=(1.67\pm0.48)\times10^7$ M$_\odot$ and effective radius $r_{\rm NSC,e}=5.0\pm1.0$~pc (or $0''.15\pm0''.03$). The mass profiles of the CMC and NSC are well described by Sérsic functions with indices $1-1.4$. Our $M_{\rm BH}$ and $M_{\rm NSC}$ estimates for NGC 3593 agree well with the recently compiled $M_{\rm BH}$-$M_{\rm NSC}$ scaling relation. Although the $M_{\rm NSC}$ uncertainty is twice the inferred $M_{\rm BH}$, the rapid central rise of the rotation velocities of the CND (as the radius decreases) clearly suggests a SMBH. Indeed, our dynamical models show that even if $M_{\rm NSC}$ is at the upper end of its allowed range, the evidence for a black hole does not vanish, but remains with a lower limit of $M_{\rm BH}>3\times10^5$ M$_\odot$.