Rocksteady is currently telling players that Batman: Arkham City should be temporarily run using DX 9 rather than DX 11. Apparently DX 11 is causing performance issues:
We have received reports of performance issues from players of Batman: Arkham City on PC. After researching the matter, we found that running the game with DX 11 is causing the performance issues. We
Batman: Arkham City PC DirectX 11 Issues