[Math] Algebraic Geometry versus Complex Geometry

ag.algebraic-geometrycomplex-geometry

This question is motivated by this one.

I would like to hear about results concerning complex projective varieties which

  1. have a complex analytic proof but no known algebraic proof; or
  2. 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?

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