Q1: How do I prove $h^i(X,\Omega^j_X)=h^j(X,\Omega^i_X)$ for say $X$ smooth geometrically integral projective variety over some field of characteristic zero (I have a strong suspicion this should be true, maybe after changing hypotheses a bit)?
Q2: More vaguely, suppose I have some purely geometric result I see somewhere proved using hodge-theoretic methods, and I want to prove it without using hodge theory, what are the tools I'll have to use to do that?
Q2': For example, is there an algebraically defined cohomology theory for smooth projective varieties that carries hodge structures? (I suspect no, but then why not and what do we do about it?)