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Aug 3, 2014 at 7:19 vote accept Dubious
Jul 30, 2014 at 7:18 comment added Ariyan Javanpeykar @DanielLoughran Your guess is correct. See Criterion 1, page 3, of Gonzalez-Diez`s paper citeseerx.ist.psu.edu/viewdoc/… .
Jul 29, 2014 at 16:43 history edited Daniel Loughran CC BY-SA 3.0
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Jul 29, 2014 at 16:28 comment added Daniel Loughran Yes you are quite right... I shall edit my answer accordingly.
Jul 29, 2014 at 15:30 comment added Matthias Wendt I think that a complex surface would always be definable over a finitely generated extension of $\mathbb{C}$, so your guess would imply that there are always only countably many isomorphism classes.
Jul 29, 2014 at 15:18 history answered Daniel Loughran CC BY-SA 3.0