  We investigate the effects of AGN heating and the ultraviolet background on
the low-redshift Lyman-$\alpha$ forest column density distribution (CDD) using
the Illustris simulation. We show that Illustris reproduces observations at $z
=0.1$ in the column density range $10^{12.5} - 10^{13.5}$cm$^{-2}$, relevant
for the "photon underproduction crisis." We attribute this to the inclusion of
AGN feedback, which changes the gas distribution so as to mimic the effect of
extra photons, as well as the use of the Faucher-Gigu\`ere ultraviolet
background, which is more ionizing at $z=0.1$ than the Haardt & Madau
background previously considered. We show that the difference between
simulations run with smoothed particle hydrodynamics and simulations using a
moving mesh is small in this column density range but can be more significant
at larger column densities. We further consider the effect of supernova
feedback, Voigt profile fitting and finite resolution, all of which we show to
have little influence on the CDD. Finally, we identify a discrepancy between
our simulations and observations at column densities $10^{14} -
10^{16}$cm$^{-2}$, where Illustris produces too few absorbers, which suggests
the AGN feedback model should be further refined. Since the "photon
underproduction crisis" primarily affects lower column density systems, we
conclude that AGN feedback and standard ionizing background models can resolve
the crisis.
