evidence: The booking photo was taken on August 24, when Trump was arrested in Fulton County, Georgia, on charges connected to his efforts to overturn his defeat in the state in the 2020 election. [...] Facts First: National public polls do not show anything close to an increase of "four and five times" in Black support for Trump since his mug shot was taken, either in a race against President Joe Biden or in his own favorability rating; Trump’s campaign did not respond to CNN’s request to identify any poll that corroborates Trump’s claim. Most polls conducted after the release of the mug shot did find a higher level of Black support for Trump than he had in previous polls – but the increases were within the polls’ margins of error, not massive spikes, so it’s not clear whether there was a genuine improvement or the bump was just statistical noise. In addition, one poll found a decline in Trump’s strength with Black voters in a race against Biden, while another found a decline in his favorability with Black respondents even as he improved in a race against Biden. [...] One of the five polls, by Emerson College, showed Trump’s standing with Black registered voters worsening after the mug shot was released, though this change was also within the margin of error. In Emerson’s mid-August poll, Trump had about 27% Black support in a race against Biden; in its late-August poll, he had about 19% support.