I am learning about complex algebraic varieties and have encountered several equivalence relations: isomorphism, birational, deformation equivalent, diffeomorphic and homeomorphic. I consider all of these equivalence relations with respect to the analytic topology (but I only consider algebraic varieties).
First question: Are there any other equivalence relations worth knowing?
What are the correct implications? Of course "isomorphic" implies "everything". "Diffeomorphic" implies "homeomorphic", but what is the relation between "deformation equivalent" and "diffeomorphic" for instance?
What are some standard examples and where can I read about them?
I would like to emphasize that I am learning about smooth projective varieties over the complex numbers. But certainly these notions also appear in the study of real manifolds, right?
Note. Varieties X and Y are deformation equivalent if there is a connected space T and a family of varieties over T such that X and Y arise as fibres of this family. I also want this family to be algebraic in general.