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