The PC version of Batman: Arkham City has suffered a bit of a shakey start as the title does not seem to cooperate with DirectX 11. Rocksteady has now addressed the issue on their forums.
When running Arkham City alongside DX 11, this will cause some performance issues, however, these can be avoided by running the game with DX 9 instead. Rocksteady have listed ways to disable DX 11 whether you are using Windows Live or Steam.
Community manager Andy Cataldo also added: "We sincerely apologize for any inconvenience with your gameplay experience and thank you all for your patience as we work to resolve this issue."
Rocksteady did not give a clear answer on when they will be able to have the DX 11 issue solved, but promise that it will be in the near future. If you're dying to play Arkham City though, switching to DX 9 may be your best bet.
Source: Batman: Arkham City Forums