1. To allow something to remain in a raised or higher position. A noun or pronoun can be used between "leave" and "up." Do you mind if I leave up a couple of these flyers around your store? I like the idea of a convertible, but I always end up leaving the top up because I hate having wind blowing in my face while I drive.
2. To go without taking someone or something from a lower position, place, or level. A noun or pronoun can be used between "leave" and "up." I left the kids up in their bedrooms while I went down to do the dishes. I think I left my phone up in the cabin at the lake.
3. To cause one to be in a better financial level than before. A noun or pronoun can be used between "leave" and "up." The huge market success of our latest product left us up by about $2.5 million for the quarter. The market surge is expected to leave investors up as much as 4.5 per cent.
4. To remain cautious, vigilant, guarded, et.. A noun or pronoun can be used between "leave" and "up." Tom isn't a very trusting guy, and he tends to leave his guard up even around relatively close friends. There will always be people in this business looking to exploit you, so you've got to always leave your defenses up.
leave something up
to leave something in a raised or high position. It's still warm in here. Please leave the window up. Please leave up the window. It's so hot!