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.