  We present CO(1-0) maps of 12 warm H$_2$-selected Hickson Compact Groups
(HCGs), covering 14 individually imaged warm H$_2$ bright galaxies, with CARMA.
We found a variety of molecular gas distributions within the HCGs, including
regularly rotating disks, bars, rings, tidal tails, and possibly nuclear
outflows, though the molecular gas morphologies are more consistent with
spirals and early-type galaxies than mergers and interacting systems. Our
CO-imaged HCG galaxies show star formation suppression of
$\langle$S$\rangle$=10$\pm$5, distributed bimodally, with five objects
exhibiting suppressions of S$\gtrsim$10 and depletion timescales
$\gtrsim$10Gyr. This star formation inefficiency is also seen in the efficiency
per freefall time. We investigate the gas-to-dust ratios of these galaxies to
determine if an incorrect conversion caused the apparent suppression and find
that HCGs have normal ratios. It is likely that the cause of the suppression in
these objects is associated with shocks injecting turbulence into the molecular
gas. Galaxies with high star formation suppression (S$\gtrsim$10) also appear
to be those in the most advanced stages of transition across optical and
infrared color space. This supports the idea that some galaxies in HCGs are
transitioning objects, where a disruption of the existing molecular gas in the
system suppresses star formation by inhibiting the molecular gas from
collapsing and forming stars efficiently. These observations, combined with
recent work on poststarburst galaxies with molecular reservoirs, indicates that
galaxies do not need to expel their molecular reservoirs prior to quenching
star formation and transitioning from blue spirals to red early-type galaxies.
This may imply that star formation quenching can occur without the need to
starve a galaxy of cold gas first.
