Wood glue of any type aliphatic glues like titebond seem to be popular in the.
Split wood door hinge.
Use wood putty to fill the crack and paint or varnish the door as needed.
For a more serious split you may be able to glue and clamp the door.
A minor split may be cosmetic.
It may have holes for screw mounting or spots for welding depending on the style of the hinge and the door material it is mounting to.
In particular the side where the hinges fasten onto the door tends to crack or split in certain situations.
I m thinking the best way to fix this short of ripping out the whole door frame would be to drill out the split wood chisel it square then glue in a new piece of wood for the hinge to be screwed into.
How to fix a door that cracked where the hinges go.
At this point you may need to decide if you will be happy with the patched up door frame when you are done.
Serious splits in door frames.
This video shows how to glue a cracked door stile back in place.
Lift off hinges are surface or mortise mounted devices that allow users to quickly detach the mating door while keeping hardware in place.