I am current studying Calabi-Yau theorem and some consequences. One of the most significant results is the one states every Calabi-Yau manifold is projective.

Another interesting result, if I am not mistaken, is the one proved by Nash that states smooth manifolds admit an embbeding on algebraic varieties.

My question is: is possible to determine informations of the geometry and topology of manifolds by analyzing their algebraic descriptions? The fact Calabi-Yau manifolds are projective does suggest that perhaps the correct context for studying such manifolds is the algebraic one?

What do I gain knowing a complex manifold is projective?

I am so sorry if this questions is vague at all, but this is a truly real question for someone who doesn't know anything about algebraic geometry.