I'm not positive, but I don't think the Braves are paying anything. Let me dig up the deadspin article.
Here's one of them:
http://deadspin.com/the-braves-are-going-to-ridiculous-lengths-to-squeeze-e-1782862795
http://deadspin.com/cobb-county-needs-to-raise-taxes-to-pay-for-public-park-1780832038
That's crazy. That being said though, while I hate these, I would be singing a different tune if Comcast or the Phillies ownership group or Jeffrey Lurie said they were moving the team if they couldn't get funding for the stadium from the city...