  Several recent observational studies have concluded that the initial mass
function (IMF) of stars varies systematically with galaxy properties such as
velocity dispersion. In this paper, we investigate the effect of linking the
circular velocity of galaxies, as determined from the Fundamental Plane and
Tully-Fisher relations, to the slope of the IMF with parameterizations guided
by several of these studies. For each empirical relation, we generate stellar
masses of ~600,000 SDSS galaxies at z ~ 0.1, by fitting the optical photometry
to large suites of synthetic stellar populations that sample the full range of
galaxy parameters. We generate stellar mass functions and examine the
stellar-to-halo mass relations using sub-halo abundance matching. At the
massive end, the stellar mass functions become a power law, instead of the
familiar exponential decline. As a result, it is a generic feature of these
models that the central galaxy stellar-to-halo mass relation is significantly
flatter at high masses (slope ~ -0.3 to -0.4) than in the case of a universal
IMF (slope ~ -0.6). We find that regardless of whether the IMF varies
systematically in all galaxies or just early types, there is still a
well-defined peak in the central stellar-to-halo mass ratio at halo masses of ~
10E12 solar masses. In general, the IMF variations explored here lead to
significantly higher integrated stellar densities if the assumed dependence on
circular velocity applies to all galaxies, including late-types; in fact the
more extreme cases can be ruled out, as they imply an unphysical situation in
which the stellar fraction exceeds the universal baryon fraction.
