I am current studying the Calabi-Yau theorem and some of its consequences. One of the most significant results is the one that states that every Calabi-Yau manifold is projective.
Another interesting result, if I am not mistaken, is the one proved by Nash that states that any smooth manifold admits an embedding as an algebraic variety.
My question is: is possible to determine information on the geometry and topology of manifolds by analysing their algebraic descriptions? The fact that Calabi-Yau manifolds are projective suggests that perhaps the correct context for studying such manifolds is the algebraic one.
What do I gain by knowing that a complex manifold is projective?
I am so sorry if this question is vague at all, but this is a truly real question for someone who doesn't know anything about algebraic geometry.