If you fell in a public area of your apartment complex, you may be able to hold your landlord liable if they failed to address a hazard they knew about or reasonably should have known about.
The important thing to know is that you can’t sue your landlord just because you fell. You have to prove that the property owner (or an employee or agent) should have recognized a dangerous condition and removed or repaired it, but failed to do so. Or, you need to demonstrate that they caused the dangerous condition (like leaving objects on a stairway, for example) and you were hurt as a result.