Depends on his post-flop stats, methinks. You need to be looking at Fold to C-Bet on Flop and Turn. AQ ain't in this guys range, certainly not by the turn. AK's a long shot. Your own stats at the table also come into account, I think. If you've been 3-betting like mad then he might not be so scared of you on the flop. Looks like you're pretty committed though, the most reasonable chance of him having an ace is him having bullets. Trouble is, he might be slowplaying a set. If you're ahead, you check, he checks, or you bet, he folds, no difference. If you're behind and you check, he goes all in and you dunno what to do, if you're behind and you push, obviously you're getting called. If he never seems to call down much (high FTCB on flop and turn, high AF/AFq), then he is perhaps slowplaying a PP that hit a set... By the same token, he could have JJ... 6 combos of JJ, 3 combos of QQ and TT and AA... 1 combo of KK... JJ seems the most likely given his actions and the 'combinatorics' or whatever they call it...
I think the bottom line is that you're pretty committed here. You beat 6 combos, get beat by 9 combos, and split with 1 combos, so you're behind his range, but only at a 3:2 ratio, while the pot is giving you 3:1. Check-call - if he has jacks and you have him beat, he'll likely be pushing on that river ace.
That's my analysis anyway, I'm not the most expert member here, so see what others think.
EDIT: Sorry I typed that while I was thinking it out, the first bit about flop/turn stats is pretty pointless, it's much more a question of his range at the end of the hand. My 2nd paragraph is a lot more relevant than the first, the first was me sort of thinking it through, lol.