This question is motivated by this one.
I would like to hear about results concerning complex projective varieties which
- have a complex analytic proof but no known algebraic proof; or
- have an algebraic proof but no known complex analytic proof.
For example, I don't think there exists an equivalent of Mori's bend-and-break argument that avoids reduction to positive characteristic. So the existence of rational curves on Fano varieties would be an example of 2.
Best Answer
Here is one I am curious about : Suppose X is a proper variety over $\mathbb{C}$. Then there are only finitely etale covers of X in each degree.
This is proven in SGA 1 by comparison with the classical fundamental group, but is there a purely algebraic proof?