I don't own Sleepy Hollow and never will. No copyright infringement intended.
Ichabod Crane has a huge problem (one that isn't the really big issue he and Abbie share) – He's in love with his partner. And he can't be feeling this way for her. Katrina is still a presence in his life and Ichabod feels like he's betraying her. He's never stopped loving her and this is something that is unforgivable. So he suppresses his feelings for Abbie and pretends like nothing's there. Everything is fine. He's even convinced himself of that - It's worked so far and Ichabod hopes it will continue to do so.
Besides, he and Abbie have more important things to do and don't have time to worry about their love lives (or lack thereof). The apocalypse takes precedence over everything. And this new technology he's still learning about is fascinating, even though Abbie gets frustrated teaching him about it (he feels guilty about that sometimes, but is trying to "get over it").
Solving Sleepy Hollow's crimes are a nice distraction too, and Ichabod takes advantage of it so he doesn't have to think about anything he doesn't want to (it's extremely hard, however, when he and Abbie spend so much time together).
Katrina is the love of Ichabod's life and he needs to remember that. She would be furious if she discovers the truth, and he doesn't want to lose their connection. They promised to always be faithful and it feels like he's doing the opposite of that even though nothing has (nor will ever) happened. He sometimes worried that his wife knows, but if she does, she's never said anything to him. He definitely doesn't plan on saying anything to her. That would end horrifically and Ichabod has no plans to hurt her. What if she hurts Abbie? Or does something else? He can't do that to her.
Abbie sometimes notices when he acts odd around her, but she just rolls her eyes and ignores him because she's used to it. He still hasn't adjusted to this time period, although he likes the various places she's dragged him to just to see his reactions. Most of the food Abbie's made him try has been amazing and he enjoys it.
Ichabod's interrupted in his thoughts by Abbie. "You've become my best friend, you know that?"
His heart breaks, but he pastes a smile on his face. "And you are mine."
And that's all they can ever be.