  In several previous studies, quasars exhibiting broad emission lines with
>1000 km/s velocity offsets with respect to the host galaxy rest frame have
been discovered. One leading hypothesis for the origin of these velocity-offset
broad lines is the dynamics of a binary supermassive black hole (SMBH). We
present high-resolution radio imaging of 34 quasars showing these
velocity-offset broad lines with the Very Long Baseline Array (VLBA), aimed at
finding evidence for the putative binary SMBHs (such as dual radio cores), and
testing the competing physical models. We detect exactly half of the target
sample from our VLBA imaging, after implementing a 5 detection limit. While we
do not resolve double radio sources in any of the targets, we obtain limits on
the instantaneous projected separations of a radio-emitting binary for all of
the detected sources under the assumption that a binary still exists within our
VLBA angular resolution limits. We also assess the likelihood that a
radio-emitting companion SMBH exists outside of our angular resolution limits,
but its radio luminosity is too weak to produce a detectable signal in the VLBA
data. Additionally, we compare the precise sky positions afforded by these data
to optical positions from both the SDSS and Gaia DR2 source catalogs. We find
projected radio/optical separations on the order of 10 pc for three quasars.
Finally, we explore how future multi-wavelength campaigns with optical, radio,
and X-ray observatories can help discriminate further between the competing
physical models.
