Light was not the faithful type, and is certainly not the best boyfriend. But hey, at least he does love you, doesn't he?
He would constantly always be like don't do this, don't go there, block them, stop talking with them, etc. Light wasn't the jealous type, but is actually more possessive and very controlling. And him having god complex doesn't make it better. And he's definitely not letting you go, he believes that you're the one for him in this rotten world.
Mind you, he's also secretive. Light doesn't have that much to hide from you. But there's this one thing. He would do anything he can to not let you find out that he's Kira. Or else, he knows you would've left him and he'll have to do something about it. That's another story to worry about.
Light sat at his desk with a natural expression as he does his homework while you laid on his bed, waiting for him to finish.