WISDOM Project -- XXVI. Cross-checking supermassive black hole mass estimates from ALMA CO gas kinematics and SINFONI stellar kinematics in the galaxy NGC 4751

We present high angular resolution ($0.19''$ or $\approx24$ pc) ALMA observations of the $^{12}$CO(3-2) line emission of the galaxy NGC 4751. The data provide evidence for the presence of a central SMBH. Assuming a constant mass-to-light ratio ($M/L$), we infer a SMBH mass $M_\text{BH}=3.43^{+0.45}_{-0.44}[\text{stat},3σ]^{+0.22}_{-0.64}[\text{sys}]\times10^9$ M$_\odot$ and a F160W filter stellar $M/L_{F160W}=2.68\pm0.11[\text{stat},3σ]^{+0.10}_{-0.80}[\text{sys}]$ M$_\odot$/L$_{\odot,\text{F160W}}$, where the first uncertainties are statistical and the second systematic. Assuming a linearly spatially-varying $M/L$, we infer $M_\text{BH}=2.79^{+0.75}_{-0.57}[\text{stat},3σ]^{+0.75}_{-0.45}[\text{syst}]\times10^9$ M$_\odot$ and $(M/L_\text{F160W})/(\text{M}_\odot/\text{L}_{\odot,\text{F160W}})=3.07^{+0.27}_{-0.35}[\text{stat},3σ]^{+0.08}_{-1.14}[\text{sys}]-0.09^{+0.08}_{-0.06}[\text{stat},3σ]^{+0.08}_{-0.01}[\text{sys}](R/\text{arcsec})$, where $R$ is the galactocentric radius. We also present SMBH mass estimates using the Jeans Anisotropic Modelling (JAM) method and Very Large Telescope Spectrograph for INtegral Field Observations in the Near Infrared (SINFONI) stellar kinematics. Assuming a cylindrically-aligned velocity ellipsoid (JAM$_\text{cyl}$) we infer $M_\text{BH}=(2.52\pm 0.36)\times10^9$ M$_\odot$, while assuming a spherically-aligned velocity ellipsoid (JAM$_\text{sph}$) we infer $M_\text{BH}=(3.24\pm0.87)\times10^9$ M$_\odot$. The SMBH mass assuming a constant $M/L$ is statistically consistent with that of JAM$_\text{sph}$, whereas the mass assuming a linearly-varying $M/L$ is consistent with both JAM$_\text{cyl}$ and JAM$_\text{sph}$ (within the uncertainties). Our derived masses are larger than (and inconsistent with) one previous stellar dynamical measurement using the Schwarzschild orbit-superposition method and the same SINFONI kinematics.