The complexity of the classification of Riemann surfaces and complex manifolds

In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of $\mathbb{C}$ can be used as complete invariants for Riemann surfaces considered up to conformal equivalence, and that this equivalence relation is itself 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 modem theory of Borel equivalence relations. ¶ On the other hand we show that the analog of Becker, Rubel, and Henson's question has a negative solution in (complex) dimension $n \geq 2$.