What is the definition of sheaf of meromorphic differential form on a complex manifold？

Question

Let $(X,\mathcal{O}_X)$ be a complex manifold where $\mathcal{O}_X$ is the sheaf of $\mathbb{C}$-valued holomorphic functions on $X$.

Let $T_X^\vee$ be the cotangent bundle over $X$ with the projection map $p:T_X^\vee\to X$. Let $\Omega^1_X$ be the sheaf on $X$ induced by $T_X^\vee$: $$U\mapsto \{f:U\to T_U^\vee :\text{sections of }p \text{ over }U, \text{i.e. }p\circ f\equiv \mathrm{id}_U\}$$ We also call $\Omega^1_X$ the sheaf of 1-differential forms on $X$, or the sheaf of holomorphic differential forms on $X$.

Let $\mathcal{M}_X$ be the sheaf of meromorphic functions on $X$, i.e. $\mathcal{M}_X(U)=\{\text{meromorphic functions }f:U\to\mathbb{P}^1\}$

Now can we define the sheaf of meromorphic differential form on $X$ as $\mathcal{M}_X\otimes_{\mathcal{O}_X}\Omega^1_X$?

A reference with this definition will be nice.

Answer 1

Yes, that is a good definition. In general if $\mathcal{V}$ is a vector bundle then we can define the sheaf of meromorphic sections of $\mathcal{V}$ analogously.