Prospects for Measuring the Mass of Black Holes at High Redshifts with
  Resolved Kinematics Using Gravitational Lensing
  Application of the most robust method of measuring black hole masses,
spatially resolved kinematics of gas and stars, is presently limited to nearby
galaxies. The Atacama Large Millimeter/sub-millimeter Array (ALMA) and thirty
meter class telescopes (the Thirty Meter Telescope, the Giant Magellan
Telescope, and the European Extremely Large Telescope) with milli-arcsecond
resolution are expected to extend such measurements to larger distances. Here,
we study the possibility of exploiting the angular magnification provided by
strong gravitational lensing to measure black hole masses at high redshifts (z~
1-6), using resolved gas kinematics with these instruments. We show that in
~15% and ~20% of strongly lensed galaxies, the inner 25 and 50 pc could be
resolved, allowing the mass of ~$10^8 M_{\odot}$ black holes to be dynamically
measured with ALMA, if moderately bright molecular gas is present at these
small radii. Given the large number of strong lenses discovered in current
millimeter surveys and future optical surveys, this fraction could constitute a
statistically significant population for studying the evolution of the
M-$\sigma$ relation at high redshifts.
