Let $k$ be an algebraically closed field of characteristic 0.

I am wondering if there are proofs of the following facts that do not use the analytic topology over $\mathbb{C}$:

1. Let $X$ be a smooth projective variety over $k$. Then $h^{p,q}(X)=h^{q,p}(X)$ for all $p,q$.
2. Let $f:X\to Y$ be a smooth proper morphism of varieties over $k$. Then $h^{p,q}(X_y)$ is locally constant on $Y$.

Unlike other facts about Hodge numbers, these are only true over fields of characteristic 0.

More generally, I was wondering if there is a purely algebraic way (i.e. avoiding the analytic topology) of "doing Hodge Theory."