I'm reading the book "Principles of Algebraic Geometry". After proving the statement


*Every meromorphic function on an algebraic variety $V \subset \mathbb{P}^n$ is rational*


Griffith and Harris say that it is not hard to see(p 170):


1. Any meromorphic differential form on a smooth variety is algebraic.
2. Any holomorphic map between smooth varieties may be given by rational functions.
3. $\cdots$


They also give the proof for 1 and 2. For the first statement, I can't understand why the coefficients are rational(they say:  *$\cdots$ of these forms with meromorphic, hence rational, coefficient functions*). I think the key point is that the meromorphic differential is globally defined, but I can't see the role of the condition in the proof. For the second one, how to deduce the map is rational from the graph $\Gamma$ is algebraic? Thank you for any comment!



Application of the G.A.G.A. principle
I'll address the question about rational differential forms, since that one is a bit more subtle (for reasons indicated below). As you note, the global nature of meromorphicity and compactness need to be used in some way (i.e., if we merely analytify the sheaf of rational functions we do **not** get the sheaf of meromorphic functions). The key is to introduce certain global coherent ideal sheaves of denominators and to exploit that depth of a finitely generated module over a noetherian local ring can be computed on completions. To clarify the ideas, for much of what follows we avoid all smoothness hypotheses.


Let's first make some purely analytic definitions having nothing to do with algebraic geometry or global compactness. For any complex-analytic space $X$, define the sheaf $M\_X$ of *meromorphic functions* to be the localization of $O\_X$ at the multiplicative subsheaf $S\_X \subset O\_X$ of regular local section (i.e. $S\_X(U)$ is the set of $s \in O\_X(U)$ for which $s$-multiplication on $O\_U$ is injective). In an evident manner $O\_X \subset M\_X$ as sheaves of rings. For every $f \in M\_X(U)$, the ideal sheaf $D\_f \subset O\_X|\_U$ of "denominators" of $f$ is defined as follows: $D\_f(V)$ is the set of $s \in O\_X(V)$ such that $sf \in O\_X(V)$ inside $M\_X(V)$ for all open $V \subset U$. It is elementary to check that $D\_f$ is coherent and locally on $U$ contains a section of $S\_X$. 


The set of coherent ideal sheaves $I$ on $X$ which locally contain a section from $S\_X$ is directed by reverse inclusion (if $I$ and $J$ are two such, $IJ$ is a third). The two key points about such $I$ are as follows:


(i) A coherent ideal $I$ is of this type if and only if each stalk $I\_x$ has positive depth over the local noetherian ring $O\_x$, a property that is equivalent to the same for completed stalks. In particular, if $X = Y^{\rm{an}}$ for a locally finite type $\mathbf{C}$-scheme $Y$ and $I=J^{\rm{an}}$ for a coherent ideal sheaf $J$ on $Y$ then $I$ has positive depth for all of its stalks if and only if the same holds for $J$ (as for the latter we only need to check at closed points).


(ii) For any such $I$ we have naturally via the link to sections of $S\_X$ that $${\rm{Hom}}\_X(I, O\_X) \subset M\_X(X)$$ (check!).


By (ii) we may pass to the direct limit over all such $I$ to obtain $$\varinjlim {\rm{Hom}}\_X(I,O\_X) \subset M\_X(X),$$
and **this is an equality** since for any $f \in M\_X(X)$ we see that $f$-multiplication defines an element of ${\rm{Hom}}\_X(D\_f, O\_X)$ that is carried to $f$. (Note that we are not assuming $X$ to be compact.)


The same procedures apply to global sections of the sheaf $M\_X \otimes E$ for any vector bundle $E$ on $X$; we call this sheaf the "sheaf of meromorphic sections of $E$", and its sections over any open $U$ do indeed match the usual such notion. Taking $E = \Omega^j\_X$ for smooth $X$ thereby recovers the usual notion of sheaf of meromorphic $j$-forms, and 
$$\varinjlim {\rm{Hom}}\_X(I, E) = (M\_X \otimes E)(X)$$
for any $E$ (such as $E = \Omega^j\_X$ when $X$ is smooth).


Now suppose $X = Y^{\rm{an}}$ for a locally finite type $\mathbf{C}$-scheme $Y$ and $E = F^{\rm{an}}$ for a vector bundle $F$ on $Y$; e.g., if $X$ is smooth and $E = \Omega^j\_X$ then $Y$ is smooth and we can take $F = \Omega^j\_{Y/\mathbf{C}}$. For any coherent ideal $J$ on $Y$ with positive depth at all stalks we see that the coherent ideal $J^{\rm{an}}$ on $X$ has positive depth at all stalks, so there is an evident inclusion
$${\rm{Hom}}\_Y(J, F) \hookrightarrow {\rm{Hom}}\_X(J^{\rm{an}}, E)$$
and passing to the limit over all such $J$ and realizing the collection of $J^{\rm{an}}$'s as a subcollection of that of $I$'s above recovers the usual inclusion $(M\_Y \otimes F)(Y) \rightarrow (M\_X \otimes E)(X)$ induced by the sheaf inclusion $(M\_Y)^{\rm{an}} \subset M\_X$. This latter inclusion of sheaves is **never** an equality in dimension $> 1$ (e.g., it fails even for the projective plane) due to local divisors not arising from algebraic ones when the divisors aren't points, and that underlies the subtlety of the matter because despite the failure of such equality for sheaves we will nonetheless get equality of global sections in the proper case due to GAGA.


So now we can finally bring in GAGA: assume $X = Y^{\rm{an}}$ for proper $Y$, and let $F$ be a vector bundle on $Y$ (such as $F = \Omega^j\_{Y/\mathbf{C}}$ for smooth $Y$). Then by GAGA we have a bijection between the sets of coherent sheaves of ideals on $X$ and $Y$, and by (i) above this bijection preserves the condition "positive depth for stalks at all points" because we can compute depth on completed stalks! Consequently, the natural map
$$\varinjlim {\rm{Hom}}\_Y(J, F) \rightarrow \varinjlim {\rm{Hom}}\_X(I, F^{\rm{an}})$$
(with $J$ and $I$ respectively varying through the directed systems of coherent ideal sheaves on $Y$ and $X$ with positive depth at all stalks) is an *equality*, so this identifies the natural inclusion
$$(M\_Y \otimes F)(Y) \hookrightarrow (M\_X \otimes F^{\rm{an}})(X)$$
with an equality. Taking $F = \Omega^j\_{Y/\mathbf{C}}$ for smooth $Y$ then answers the question posed.


