Algebraic Geometry versus Complex Geometry

I am fairly certain that there is no complex analytic proof of the following theorem (but I would love to be proven wrong!). This is not strictly speaking an answer to the question, because the available proof is not exactly algebraic either; rather, it uses $p$-adic (analytic) methods.

Theorem. (Batyrev) Let $X$ and $Y$ be birational Calabi–Yau varieties (that is, smooth projective over $mathbb C$ with $Omega^n cong mathcal O$). Then $H^i(X,mathbb C) cong H^i(Y,mathbb C)$.

The same methods were later refined to prove the following theorem:

Theorem. (Ito) Let $X$ and $Y$ be birational smooth minimal models (that is, smooth projective over $mathbb C$ with $Omega^n$ nef). Then $h^{p,q}(X) = h^{p,q}(Y)$ for all $p,q$.

Again, the proof goes through $p$-adic analytic methods, this time combined with $p$-adic Hodge theory (which I think counts as an algebraic method).

References.

V. V. Batyrev, Birational Calabi–Yau $n$-folds have equal Betti numbers. arXiv:alg-geom/9710020

T. Ito, Birational smooth minimal models have equal Hodge numbers in all dimensions. arXiv:math/0209269