On page 78 of Hartshorne's "Algebraic Geometry," we encounter the following correspondence:
Proposition 2.6: Let $k$ be an algebraically closed field. There exists a natural, fully faithful functor $t: \mathfrak{Var}(k) \rightarrow \mathfrak{Sch}(k)$ from the category of varieties over $k$ to schemes over $k$. For any variety $V$, its topological space is homeomorphic to the set of closed points of $\operatorname{sp}(t(V))$ (where $\operatorname{sp}(t(V))$ denotes the underlying topological space), and its sheaf of regular functions is obtained by restricting the structure sheaf of $t(V)$ via this homeomorphism.
In other words, one can associate a scheme to a variety since $t$ is faithful.
Now, let $k=\mathbb{C}$. Then one can also associate to a scheme a complex analytic variety, via the well-known GAGA theorem. GAGA also tells us that many things are equivalent between the scheme category and the complex analytic category; in fact, the correspondence between them is a fully faithful functor. We denote this functor by $G$. (I believe $t^{-1}$ aligns with the GAGA correspondence, as suggested by references on page 439 and page 81-2.14-d in Hartshorne book; however, this is not the focus of this post.)
However, several important questions arise, which may require additional conditions on the involved schemes:
Question:
Is $H^i(X, F)\cong H^i(G(X), G(F))$ true, where $F$ is some good sheaf on the scheme $X$?
If $X$ is birational to $Y$, then is $G(X)$ birational (or bimeromorphic) to $G(Y)$?
Suppose $f: X\to Y$ is a morphism of schemes, and the generic fiber $X_{\eta}$ is rational, i.e., birational to $\mathbb{P}^n_{K}$, where $K$ is the function field of $X$. Then is the general fiber of $G(f): G(X)\to G(Y)$ birational to $\mathbb{CP}^n:=\mathbb{C}^{n+1}/\sim$, where the general fiber means the fiber over an analytic Zariski open subset of $G(Y)$.
Any comments and recommended references would be greatly appreciated.