Metallicity evolution of direct collapse black hole hosts: CR7 as a case
  study
  In this study we focus on the $z\sim6.6$ Lyman-$\alpha$ CR7 consisting of
clump A that is host to a potential direct collapse black hole (DCBH), and two
metal enriched star forming clumps B and C. In contrast to claims that
signatures of metals rule out the existence of DCBHs, we show that metal
pollution of A from star forming clumps clumps B and C is inevitable, and that
A can form a DCBH well before its metallicity exceeds the critical threshold of
$10^{-5}-10^{-6}\ \rm Z_{\odot}$. Assuming metal mixing happens
instantaneously, we derive the metallicity of A based on the star formation
history of B and C. We find that treating a final accreting black hole of
$10^6-10^7\ \rm M_{\odot}$ in A for nebular emission already pushes its
$H_{160}$ - [3.6] and [3.6]-[4.5] colours into the 3$\sigma$ limit of
observations. Hence, we show that the presence of metals in DCBH hosts is
inevitable, and that it is the coevolution of the LW radiation field and metals
originating from neighbouring galaxies that governs DCBH formation in a
neighbouring {initially} pristine atomic cooling haloes.
