The complexity of the classification of Riemann surfaces and complex manifolds

In answer to a question by Becker, Rubel, and Henson, we show that countabl e subsets of C can be used as complete invariants for Riemann surfaces cons idered up to conformal equivalence, and that this equivalence relation is i tself Borel in a natural Borel structure on the space of all such surfaces. We further proceed to precisely calculate the classification difficulty of this equivalence relation in terms of the modern theory of Borel equivalen ce relations. On the other hand we show that the analog of Becker, Rubel, and Henson's qu estion has a negative solution in (complex) dimension n greater than or equ al to 2.