Why are compact complex manifolds Liouville?

Question

I know this is true but strangely can't find references.

Also, consider the trivial $n$-bundle over any connected compact manifold, does Liouville imply that all holomorphic sections are constant?

Consider the tautological bundle over a Grassmanian manifold. Does Liouville imply that the only holomorphic section is the zero section?

Answer 1

Let $f: X \to \mathbb C$ an holomorphic map, non-constant

Then $f$ is open (standard argument using Maximum principle for example). But $f(X)$ is compact and open, and non-empty. Contradiction.

Conclusion : $f$ is a constant map.

Answer 2

$\newcommand{\Cpx}{\mathbf{C}}$Let $M$ be a connected compact holomorphic manifold.

Chern's Complex Manifolds without Potential Theory gives a proof in the first few pages that every holomorphic function on $M$ is constant.

It follows that if $M \times \Cpx^{k} \to M$ is a trivial holomorphic vector bundle over $M$, then every holomorphic section is constant: A holomorphic section $s:M \to M \times \Cpx^{k}$ followed by projection to the second factor is a holomorphic map $M \to \Cpx^{k}$, and each component function of this map is constant.

Since the tautological bundle of a Grassmannian $G_{k}(V)$ may be viewed as a holomorphic subbundle of the trivial bundle $G \times V \to G$, the only holomorphic section of the tautological bundle is identically zero: By the preceding paragraph, every holomorphic section is constant, and by definition of the tautological bundle, the value of a section must lie in every $k$-dimensional subspace of $V$, hence must be the zero vector.