[proofplan] We associate to the Cousin I datum its difference cocycle $f_{ij}=f_i-f_j$ on pairwise overlaps and verify the Čech cocycle identity on triple overlaps. A meromorphic global solution produces holomorphic correction terms whose Čech coboundary is this cocycle, so solvability implies vanishing of the associated class. Conversely, if the class vanishes, possibly after passing to a refinement of the cover, holomorphic correction terms make the local meromorphic functions agree on overlaps; the sheaf property for meromorphic functions then glues them to a global solution. The final assertion follows because $H^1(X,\mathcal{O})=0$ forces every such Čech class to vanish. [/proofplan] [step:Define the Cousin cocycle from the local meromorphic data] Let $\mathcal{O}$ denote the sheaf of holomorphic functions on $X$, and let $\mathcal{M}$ denote the sheaf of meromorphic functions on $X$. Write the open cover as $\mathcal{U}=\{U_i\}_{i\in I}$, where $I$ is an index set. A Cousin I datum on $\mathcal{U}$ is a family of meromorphic functions \begin{align*} f_i \in \mathcal{M}(U_i), \qquad i\in I, \end{align*} such that for every pair $i,j\in I$ the difference $f_i-f_j$ is holomorphic on $U_i\cap U_j$. For each ordered pair $(i,j)\in I\times I$, define \begin{align*} f_{ij}: U_i\cap U_j &\to \mathbb{C} \\ x &\mapsto f_i(x)-f_j(x), \end{align*} where the equality is interpreted in the sheaf $\mathcal{M}$ and the Cousin hypothesis ensures $f_{ij}\in\mathcal{O}(U_i\cap U_j)$. On every triple intersection $U_i\cap U_j\cap U_k$, we compute in $\mathcal{O}(U_i\cap U_j\cap U_k)$: \begin{align*} f_{ij}+f_{jk} &=(f_i-f_j)+(f_j-f_k) \\ &=f_i-f_k \\ &=f_{ik}. \end{align*} Equivalently, \begin{align*} f_{jk}-f_{ik}+f_{ij}=0. \end{align*} Thus the family $\{f_{ij}\}_{i,j\in I}$ is a Čech $1$-cocycle, so it determines a class \begin{align*} [f_{ij}]\in \check{H}^1(\mathcal{U},\mathcal{O}). \end{align*} [guided] We first turn the Cousin I datum into the object that Čech cohomology can see. The sheaf $\mathcal{O}$ assigns to each open set $V\subseteq X$ the holomorphic functions on $V$, while the sheaf $\mathcal{M}$ assigns to $V$ the meromorphic functions on $V$. Write the given cover as $\mathcal{U}=\{U_i\}_{i\in I}$. The Cousin I hypothesis says that we have local meromorphic functions \begin{align*} f_i\in \mathcal{M}(U_i),\qquad i\in I, \end{align*} and that their differences are holomorphic: \begin{align*} f_i-f_j\in \mathcal{O}(U_i\cap U_j) \end{align*} for every $i,j\in I$. This is exactly the condition needed to build a Čech cochain with values in $\mathcal{O}$ rather than merely in $\mathcal{M}$. For each ordered pair $(i,j)$, define \begin{align*} f_{ij}: U_i\cap U_j &\to \mathbb{C} \\ x &\mapsto f_i(x)-f_j(x). \end{align*} The map is holomorphic on $U_i\cap U_j$ by the defining condition of a Cousin I datum, so $f_{ij}\in\mathcal{O}(U_i\cap U_j)$. Now we verify the Čech cocycle condition. On a triple overlap $U_i\cap U_j\cap U_k$, all restrictions of $f_i$, $f_j$, and $f_k$ are meromorphic functions on the same open set, and the pairwise differences are holomorphic. Therefore \begin{align*} f_{ij}+f_{jk} &=(f_i-f_j)+(f_j-f_k) \\ &=f_i-f_k \\ &=f_{ik}. \end{align*} In the standard alternating Čech notation, this is \begin{align*} f_{jk}-f_{ik}+f_{ij}=0. \end{align*} Hence $\{f_{ij}\}_{i,j\in I}$ is a Čech $1$-cocycle with values in $\mathcal{O}$, and it defines a class \begin{align*} [f_{ij}]\in\check{H}^1(\mathcal{U},\mathcal{O}). \end{align*} [/guided] [/step] [step:Show that a global meromorphic solution makes the cocycle a coboundary] Assume the Cousin I problem is solvable. Thus there exists a meromorphic function $f\in\mathcal{M}(X)$ such that, for every $i\in I$, \begin{align*} g_i:=f-f_i\in\mathcal{O}(U_i). \end{align*} The family $\{g_i\}_{i\in I}$ is a Čech $0$-cochain in $C^0(\mathcal{U},\mathcal{O})$. On $U_i\cap U_j$, we compute \begin{align*} g_i-g_j &=(f-f_i)-(f-f_j) \\ &=f_j-f_i \\ &=-f_{ij}. \end{align*} Equivalently, if one uses the Čech coboundary convention \begin{align*} (\delta h)_{ij}=h_j-h_i \end{align*} for a $0$-cochain $h=\{h_i\}_{i\in I}$, then \begin{align*} (\delta g)_{ij}=g_j-g_i=f_{ij}. \end{align*} Therefore $\{f_{ij}\}$ is a Čech coboundary in $C^1(\mathcal{U},\mathcal{O})$, and hence \begin{align*} [f_{ij}]=0\in \check{H}^1(\mathcal{U},\mathcal{O}). \end{align*} Its image in the direct-limit Čech group $\check{H}^1(X,\mathcal{O})$ is also zero. [guided] Suppose a global solution exists. By definition, this means there is a meromorphic function $f\in\mathcal{M}(X)$ whose local principal part agrees with the prescribed datum. More concretely, for each $i\in I$, define \begin{align*} g_i:=f-f_i. \end{align*} The solvability condition says precisely that \begin{align*} g_i\in\mathcal{O}(U_i) \end{align*} for every $i$. Thus $\{g_i\}_{i\in I}$ is a Čech $0$-cochain with values in $\mathcal{O}$. We now compute its Čech coboundary. On an overlap $U_i\cap U_j$, \begin{align*} g_i-g_j &=(f-f_i)-(f-f_j) \\ &=f_j-f_i \\ &=-f_{ij}. \end{align*} With the common Čech convention \begin{align*} (\delta h)_{ij}=h_j-h_i \end{align*} for a $0$-cochain $h=\{h_i\}_{i\in I}$, this becomes \begin{align*} (\delta g)_{ij}=g_j-g_i=f_{ij}. \end{align*} So the cocycle $\{f_{ij}\}$ is the coboundary of the holomorphic $0$-cochain $\{g_i\}$. A Čech cohomology class vanishes exactly when its representative cocycle is a coboundary. Therefore \begin{align*} [f_{ij}]=0\in\check{H}^1(\mathcal{U},\mathcal{O}). \end{align*} Since $\check{H}^1(X,\mathcal{O})$ is obtained from cover-level Čech cohomology by passing to the direct limit over refinements, the image of a zero class from $\check{H}^1(\mathcal{U},\mathcal{O})$ is also zero in $\check{H}^1(X,\mathcal{O})$. [/guided] [/step] [step:Use vanishing of the Čech class to construct a global meromorphic solution] Assume that the image of $[f_{ij}]$ is zero in $\check{H}^1(X,\mathcal{O})$. By the definition of Čech cohomology over $X$ as the direct limit over open covers, there exists an open refinement $\mathcal{V}=\{V_a\}_{a\in A}$ of $\mathcal{U}$ and a refinement map \begin{align*} \rho:A&\to I \end{align*} such that $V_a\subseteq U_{\rho(a)}$ for every $a\in A$, and such that the pulled-back cocycle is a coboundary on $\mathcal{V}$. Define, for every ordered pair $(a,b)\in A\times A$, \begin{align*} F_{ab}:=f_{\rho(a)\rho(b)}\big|_{V_a\cap V_b}\in \mathcal{O}(V_a\cap V_b). \end{align*} There exist holomorphic functions $g_a\in\mathcal{O}(V_a)$ such that \begin{align*} F_{ab}=g_b-g_a \end{align*} on $V_a\cap V_b$ for all $a,b\in A$. For each $a\in A$, define a meromorphic function \begin{align*} h_a:=f_{\rho(a)}\big|_{V_a}+g_a\in \mathcal{M}(V_a). \end{align*} On $V_a\cap V_b$, \begin{align*} h_a-h_b &=\left(f_{\rho(a)}+g_a\right)-\left(f_{\rho(b)}+g_b\right) \\ &=f_{\rho(a)\rho(b)}-(g_b-g_a) \\ &=F_{ab}-F_{ab} \\ &=0. \end{align*} Hence the family $\{h_a\}_{a\in A}$ agrees on all pairwise overlaps. Since $\mathcal{M}$ is a sheaf, the gluing axiom gives a unique meromorphic function $h\in\mathcal{M}(X)$ such that \begin{align*} h\big|_{V_a}=h_a \end{align*} for every $a\in A$. Finally, fix $i\in I$ and restrict to an open set $V_a\subseteq U_i$ with $\rho(a)=i$. On $V_a$, \begin{align*} h-f_i=h_a-f_i=g_a\in\mathcal{O}(V_a). \end{align*} The sets $V_a$ with $V_a\subseteq U_i$ cover $U_i$ after replacing $\mathcal{V}$ by the common refinement $\{V_a\cap U_i\}_{(a,i)\in A\times I}$ if necessary. Since holomorphicity is local and $\mathcal{O}$ is a sheaf, it follows that \begin{align*} h-f_i\in\mathcal{O}(U_i) \end{align*} for every $i\in I$. Thus $h$ solves the Cousin I problem. [guided] Now assume the cohomology class vanishes in $\check{H}^1(X,\mathcal{O})$. The group $\check{H}^1(X,\mathcal{O})$ is the direct limit of the groups $\check{H}^1(\mathcal{W},\mathcal{O})$ over open covers $\mathcal{W}$ of $X$. Therefore saying that the image of $[f_{ij}]$ is zero means that, after refining the cover, the cocycle becomes a coboundary. Choose an open refinement $\mathcal{V}=\{V_a\}_{a\in A}$ of $\mathcal{U}$ and a refinement map \begin{align*} \rho:A&\to I \end{align*} with $V_a\subseteq U_{\rho(a)}$ for every $a\in A$. The pulled-back cocycle is the family \begin{align*} F_{ab}:=f_{\rho(a)\rho(b)}\big|_{V_a\cap V_b}\in\mathcal{O}(V_a\cap V_b). \end{align*} Since the pulled-back class is zero in $\check{H}^1(\mathcal{V},\mathcal{O})$, there is a holomorphic $0$-cochain $\{g_a\}_{a\in A}$ with $g_a\in\mathcal{O}(V_a)$ such that \begin{align*} F_{ab}=g_b-g_a \end{align*} on $V_a\cap V_b$ for every pair $a,b\in A$. The purpose of the functions $g_a$ is to correct the local meromorphic functions so that they agree on overlaps. Define \begin{align*} h_a:=f_{\rho(a)}\big|_{V_a}+g_a\in\mathcal{M}(V_a). \end{align*} This is meromorphic because $f_{\rho(a)}|_{V_a}$ is meromorphic and $g_a$ is holomorphic, hence also meromorphic. On a pairwise overlap $V_a\cap V_b$, we compute \begin{align*} h_a-h_b &=\left(f_{\rho(a)}+g_a\right)-\left(f_{\rho(b)}+g_b\right) \\ &=f_{\rho(a)\rho(b)}-(g_b-g_a) \\ &=F_{ab}-F_{ab} \\ &=0. \end{align*} Thus the functions $h_a$ agree on all overlaps. Because $\mathcal{M}$ is a sheaf, compatible meromorphic functions on an open cover glue uniquely. Hence there exists a unique meromorphic function $h\in\mathcal{M}(X)$ satisfying \begin{align*} h\big|_{V_a}=h_a \end{align*} for every $a\in A$. It remains to check that $h$ really solves the original Cousin problem, not merely a refined one. Fix $i\in I$. On any member $V_a$ of the refinement with $V_a\subseteq U_i$ and $\rho(a)=i$, we have \begin{align*} h-f_i=h_a-f_i=g_a\in\mathcal{O}(V_a). \end{align*} If the chosen refinement map does not assign every open set lying inside $U_i$ to the index $i$, replace $\mathcal{V}$ by the common refinement $\{V_a\cap U_i\}_{(a,i)\in A\times I}$. Pulling back a coboundary to a further refinement remains a coboundary, so the previous construction still applies. These refined open sets cover each $U_i$, and on each of them $h-f_i$ is holomorphic. Since holomorphicity is local and $\mathcal{O}$ is a sheaf, this implies \begin{align*} h-f_i\in\mathcal{O}(U_i). \end{align*} Thus $h$ is a global meromorphic solution to the Cousin I problem. [/guided] [/step] [step:Conclude the equivalence and the vanishing criterion] The previous two steps prove both implications: \begin{align*} \text{the Cousin I problem is solvable} \quad\Longleftrightarrow\quad [f_{ij}]=0\in\check{H}^1(X,\mathcal{O}). \end{align*} If $H^1(X,\mathcal{O})=0$, then under the standard identification of sheaf cohomology with Čech cohomology in degree $1$ for the sheaf $\mathcal{O}$ on the complex manifold $X$, every class in $\check{H}^1(X,\mathcal{O})$ is zero. Hence the class associated to every Cousin I datum vanishes, and by the equivalence just proved every Cousin I problem on $X$ is solvable. [/step]