  Compact, massive star forming galaxies at $z\sim2.5$ are thought to be
building the central regions of giant elliptical galaxies today. However, a
significant fraction of these objects were previously shown to have much
smaller H$\alpha$ line widths than expected. A possible interpretation is that
H$\alpha$ emission from their central regions, where the highest velocities are
expected, is typically obscured by dust. Here we present ALMA observations of
the CO(3-2) emission line of three compact, massive galaxies with H$\alpha$
line widths of FWHM(H$\alpha$)$\sim$125-260 km s$^{-1}$ to test this
hypothesis. Surprisingly, in all three galaxies, the CO line width is similar
to the H$\alpha$ line width: we find FWHM(CO)$\sim$165 km s$^{-1}$ for all
three galaxies whereas FWHM(CO)$\sim$450-700 km s$^{-1}$ was expected from a
simple virial estimator. These results show that the narrow H$\alpha$
linewidths of many compact massive star-forming galaxies are not due to
preferential obscuration of the highest velocity gas. An alternative
explanation for the narrow line widths is that the galaxies are disks that are
viewed nearly face-on. We suggest that there may be an inclination bias in the
size-mass plane, such that the apparent rest-frame optical sizes of face-on
galaxies are smaller than those of edge-on galaxies. Although not conclusive,
this hypothesis is supported by an observed anti-correlation between the size
and axis ratio of massive galaxies.
