Batman: Arkham City arrived on PC crazily late, and launched with a pretty much borked DirectX 11 mode. Which didn't make a lot of sense, given what a song and dance NVIDIA and Warner were making about its graphics, and if forum-hounds' apparent discovery that the release build was compiled back in September is to be believed. Finally, though, a patch has arrived which purports to fix it, for the price of 190MB. Unless you're running 32-bit Vista or 7, in which case it makes DX11 performance even worse. To the point that you're advised to only run the game in DX9 mode. Sigh. Another patch is apparently due, though.
A few other wotsits have been rejiggered too - full details below. No, they haven't removed any of the preposterous levels of DRM, sadly enough.
Performance / hitching issues have been greatly improved for running in DX 11.
An issue with players running out of Video Memory or encountering a Rendering Thread Exceptions has been address. This was primarily affecting 32-bit Operating Systems.
A progression block after defeating Ra's Al Ghul has been fixed. This was issue occurred primarily on lower end computer setups and described as Batman not readying his Reverse Batarang.
A crash that occurred when scrolling between the Character Bios and Arkham City Stories has been fixed.
A crash that occurred when selecting "Press Start" immediately when available at the Title Screen has been fixed. The issue was described as crashing/hard locking around the DLC (Downloadable Content) check.
Steam should do the auto-updating thing next time you run it. I'll give it a spin in a bit (in Win 7 64 bit, mercifully) and see if things are any better than they were.