  Empirical correlations between the masses of supermassive black holes (SMBHs)
and properties of their host galaxies are well-established. Among these is the
correlation with the flat rotation velocity of each galaxy measured either at a
large radius in its rotation curve or via a spatially-integrated emission line
width. We propose here the use of the de-projected integrated CO emission line
width as an alternative tracer of this rotation velocity, that has already been
shown useful for the Tully-Fisher (luminosity-rotation velocity) relation. We
investigate the correlation between CO line widths and SMBH masses for two
samples of galaxies with dynamical SMBH mass measurements, with respectively
spatially-resolved and unresolved CO observations. The tightest correlation is
found using the resolved sample of 24 galaxies as $\log
(M_\mathrm{BH}/\mathrm{M_\odot})=(7.5\pm0.1)+(8.5\pm0.9)[\log(W_\mathrm{50}/\sin
i \,\mathrm{km\,s}^{-1})-2.7]$, where $M_\mathrm{BH}$ is the central SMBH mass,
$W_{50}$ the full-width at half-maximum of a double-horned emission line
profile, and $i$ the inclination of the CO disc. This relation has a total
scatter of $0.6\,$dex, comparable to those of other SMBH mass correlations, and
dominated by the intrinsic scatter of $0.5\,$dex. A tight correlation is also
found between the de-projected CO line widths and the stellar velocity
dispersions averaged within one effective radius. We apply our correlation to
the COLD GASS sample to estimate the local SMBH mass function.
