Updated: The early problem was not correct. Thanks to Georges Elencwajg.
Does the following statement hold?
Let $N$ be a compact connected $n$-dimensional complex manifold. Let $X$ be the total space of the holomorphic tangent bundle of $N$ and let $R$ be the ring of holomorphic functions on $X.$ Since $X$ is connected, $R$ is a domain and denote $K$ as its quotient field. If $\text{tr. deg}_{\mathbb{C}} K \geq n,$ then $N$ is algebraic.