Cospatial Star Formation and Supermassive Black Hole Growth in $z \sim
  3$ Galaxies: Evidence for In-situ Co-evolution
  We present a sub-kpc localization of the sites of supermassive black hole
(SMBH) growth in three active galactic nuclei (AGN) at $z \sim 3$ in relation
to the regions of intense star formation in their hosts. These AGNs are
selected from Karl G. Jansky Very Large Array (VLA) and Atacama Large
Millimeter/submillimeter Array (ALMA) observations in the HUDF and COSMOS, with
the centimetric radio emission tracing both star formation and AGN, and the
sub/millimeter emission by dust tracing nearly pure star formation. We require
radio emission to be $\geqslant5\times$ more luminous than the level associated
with the sub/millimeter star formation to ensure that the radio emission is
AGN-dominated, thereby allowing localization of the AGN and star formation
independently. In all three galaxies, the AGN are located within the compact
regions of gas-rich, heavily obscured, intense nuclear star formation, with
$R_e = 0.4-1.1$ kpc and average star formation rates of $\simeq100-1200$
$M_\odot$yr$^{-1}$. If the current episode of star formation continues at such
a rate over the stellar mass doubling time of their hosts, $\simeq 0.2$ Gyr,
the newly formed stellar mass will be of the order of $10^{11}$ $M_\odot$
within the central kpc region, concurrently and cospatially with significant
growth of the SMBH. This is consistent with a picture of in-situ galactic bulge
and SMBH formation. This work demonstrates the unique complementarity of VLA
and ALMA observations to unambiguously pinpoint the locations of AGN and star
formation down to $\simeq30$ mas, corresponding to $\simeq 230$ pc at $z = 3$.
