The PC version of Batman: Arkham City has had some... teething problems since launch but, according to a post on the official forums, a patch is being readied to deal with the majority of those issues.
According to Rocksteady, the source of the game's issues is DX 11 and, as a short workaround, Rocksteady are suggesting players run the game with DX 9. According to the team, this is the easiest way to do this...
Instructions for changing DX 11 to DX 9 The Launcher can be accessed on Games for Windows LIVE and Steam to change DX 11 to DX 9 in the Settings menu. Change both DirectX 11 Features and DirectX 11 Tessellation to OFF.
Games for Windows LIVE Start > All Programs > Games > Right click on Batman: Arkham City and select Launcher > Settings 1. Open the Start Menu 2. Select All Programs and then the Games folder 3. Right click on Batman: Arkham City and select Launcher 4. Choose Settings then click on the arrow next to DX options to change
Steam Launch Steam Client > Library > Right click on Batman: Arkham City > Configure PC Options 1. Open Steam 2. Select Library 3. Right click on Batman: Arkham City and select Configure PC Options 4. Choose Settings then click on the arrow next to DX options to change
Rocksteady stated that it's "working on a title update to address this matter and expect to make it available in the near future."
Thanks PC Gamer.