Definition of meromorphic function on complex manifolds There are typically four(or more?) definition of meromorphic function on Riemann surface and three definitions of meromorphic function on complex manifolds, I want to show they are equivalent (I will not consider the additional one on Riemann surface as it's specified for that case):
Definition 1 Define the sheaf of meromorphic function to be
$$\mathcal{M}_X=\coprod_{x \in X} \mathcal{M}_{X, x}$$
with topology that open sets are union of $\{G_x/H_x\mid x\in V\}\subset \mathcal{M}_X$ with $V$ open and $G,H \in \mathcal{O}_X(V)$. Meromorphic function is the section of the sheaf above.
Definition 2 the sheaf of meromorphic function is sheafification of the following sheaf:
Let $U$ be an open subset $U = \cup U_i$ for $U_i$ connected component then we have the presheaf $$U\mapsto \Pi_i \text{ Frac}(\mathcal{O}_X(U_i))$$
it's not hard to show the sheaf in definition 1is the Etale space assciated to this presheaf.
Definition 3 Let $U \subset \mathbb{C}^n$ be open. A meromorphic function $f$ on $U$ is a function on the complement of a nowhere dense subset $S \subset U$ with the following property: There exist an open cover $U=\bigcup U_i$ and holomorphic functions $g_i, h_i: U_i \rightarrow \mathbb{C}$ with $\left.\left.h_i\right|_{U_i \backslash S} \cdot f\right|_{U_i \backslash S}=\left.g_i\right|_{U_i \backslash S}$.

I was trying to show that definition 3 are equivalent to 1 (and 2), however I don't have good idea.

I see the point is the section of the sheaf, is a continuous map from $X\to \mathcal{M}_X$, with the topology, contains all the $$[(U,G/H)]=\{G_x/H_x \mid x \in U\}$$ being open in the etale space topology.
Therefore if $s:X\to \mathcal{M}_X$ such that $s_x = G_x / H_x$ then locally for where $G$ and $H$ are defined, $s_y = G_y/H_y$ for $y\in U$(that's guaranteed by the continuity of the section). Therefore, locally the information of the section is equivalent to the information of pair $(G,H)$, definition 3 use that locally information $(G_\alpha,H_\alpha)$ to define the section.
And you see that $U_i$ in definition 3 are the sets of open subsets that the representative element $(G,H)$ are defined.
 A: I have two definitions that are pretty much the ones you mention. Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is the sheaf of $\mathbb{C}$-valued holomorphic functions on $X$.
1.$\mathcal{F}$ is a sheaf on $X$ s.t. it is sheaf associated to the presheaf
$$U\mapsto \mathcal{S}^{-1}(U)\mathcal{O}_X(U)$$
where $\mathcal{S}(U)=\{a\in\mathcal{O}_{X}(U)\ |\ \forall x\in U,a_{x}\text{ is not a zero-divisor of }\mathcal{O}_{X,x}\}$ and $\mathcal{S}$ is a subsheaf of sets of $\mathcal{O}_X$, see this stacks project page.
2.$\mathcal G$ is a sheaf on $X$ s.t.
$$U\mapsto\{\text{meromorphic functions on }U\}$$
where a meromorphic function on $U$ (see A course in complex analysis by Fischer, Wolfgang and Lieb, Ingo) is a holomorphic function $f:U\to \mathbb{P}^1$ of complex manifolds s.t. there exists an open covering $\{U_i,i\in I\}$ of $U$, $g_i,h_i\in \mathcal{O}_X(U_i)$ s.t.
(1) for all $i$, $h_i$ is nowhere $\equiv 0$, i.e. $h_i|_V\not\equiv 0$ for all non-empty open subset $V\subset U_i$.
(2) $g_ih_j=g_jh_i$ over $U_i\cap U_j$
(3)for all $i$, $f|_{U_i}=\frac{g_i}{h_i}:U_i \to \mathbb{P}^1$.
A non-rigorous proof is as following:

*

*Let $U$ be non-empty open in $X$. We know $$\mathcal{G}(U)=\{\text{meromorphic functions }f:U\to\mathbb{P}_{\mathbb{C}}^{1}\}.$$


*We know \begin{align}
\mathcal{F}(U)= & \{(\frac{g_{x}}{h_{x}},U_{x})_{x\in U}\in\prod_{x\in U}\mathcal{F}_{x}:\text{for all }x\in U\text{, there} \\
 & \text{exists an open neighborhood }V\subset U\text{ of }x\text{, and} \\ 
 & \frac{g_{V}}{h_{V}}\in\mathcal{F}(V)\text{ s.t. for all }y\in V, \\
 & (\frac{g_{y}}{h_{y}},U_{y})=(\frac{g_{V}}{h_{V}},V)\in\mathcal{F}_{y}\}.
\end{align}


*Let $f\in\mathcal{G}(U)$, then there exists an open covering $(U_{i})_{i\in I}$ of $U$ and $g_{i},h_{i}\in\mathcal{O}_{X}(U_{i})$ s.t. $h_{i}$ nowhere $\equiv0$, $g_{i}h_{j}=g_{j}h_{i}$ on $U_{ij}=U_{i}\cap U_{j}$ and $f|_{U_{i}}\equiv\frac{g_{i}}{h_{i}}$.


*For each $x\in U$, there exists $i_{x}\in I$ s.t. $x\in U_{i_x}$.


*For each $i\in I$, $h_{i}$ is nowhere $\equiv0$ so $h_{i}$ is not $\equiv0$ over any non-empty open subset of $U_{i}$. So for all $x\in U_i$, $(h_{i},U_{i})\neq0$ in the stalk $\mathcal{O}_{x}$, hence it is not a zero-divisor and thus $h_{i}\in \mathcal{S}(U_i)$ (the only zero-divisor is zero since the stalk is an integral domain, check the identity theorem on multivariable complex analysis, see Introduction to complex analysis in several variables by Scheidemann, Volker).


*So we have an element $(\frac{g_{i_{x}}}{h_{i_{x}}},U_{i_{x}})_{x\in U}\in\prod_{x\in U}\mathcal{F}_{x}$.


*Let $x\in U$, then for all $y\in U_{i_{x}}$, we have $(\frac{g_{i_{y}}}{h_{i_{y}}},U_{i_{y}})=(\frac{g_{i_{x}}}{h_{i_{x}}},U_{i_{x}})$ in $\mathcal{F}_{y}$. Thus $(\frac{g_{i_{x}}}{h_{i_{x}}},U_{i_{x}})_{x\in U}\in\mathcal{F}(U)$.


*Reversely given an element $(s_{x})_{x\in U}\in\mathcal{F}(U)$. For each $x\in U$, we can find an open neighborhood $U_{x}\subset U$ of $x$ and $\frac{g_{x}}{h_{x}}\in\mathcal{S}(U_{x})^{-1}\mathcal{O}(U_{x})$ s.t. for all $y\in U_{x}$, $(\frac{g_{x}}{h_{x}},U_{x})=s_{y}$.


*Then $(U_{x})_{x\in U}$ clearly forms an open covering of $U$.


*Fix $x\in U$, by definition for all $y\in U_{x}$, $(h_{x},U_{x})\neq0\in\mathcal{O}_{y}$.


*Assume for a contradiction that $h_{x}$ is not nowhere $\equiv0$, then $h_{x}\equiv0$ over some non-empty open subset of $U_{x}$, so it descends to zero over some stalk in $U_{x}$, contradiction.


*Fix $x,y\in U$, if $U_{x}\cap U_{y}\neq\emptyset$, then for any $z\in U_{x}\cap U_{y}$, we have $s_{z}=(\frac{g_{x}}{h_{x}},U_{x})=(\frac{g_{y}}{h_{y}},U_{y})\in\mathcal{F}_{z}$.


*So there exists $W\subset U_{x}\cap U_{y}$ s.t. $g_{x}h_{y}=g_{y}h_{x}$ over $W$. In particular $g_{x}(z)h_{y}(z)=g_{y}(z)h_{x}(z)$. Hence they agree over $U_{x}\cap U_{y}$.


*Hence the data glues to a holomorphic map $f:U\to\mathbb{P}_{\mathbb{C}}^{1}$ s.t. $f|_{U_{x}}=\frac{g_{x}}{h_{x}}$ for all $x\in U$.


*It can be shown that the two operations are inverse to each other, well defined (i.e. independent of choices of all those data) and preserve the $\mathbb{C}$-algebra structure, and they are compatible with restriction maps. We omit the details.


*It follows that the two definitions of sheaf of meromorphic functions on $X$ agree.
