Holomorphic functions on compact complex manifolds are constant

Remarks

Proof idea: since $X$ is compact, $\abs{f}$ attains its maximum at some point $p$. In a connected chart around $p$, the maximum modulus principle in several variables shows that $f$ equals $f(p)$ near $p$. The set of points near which $f \equiv f(p)$ is open, and by the identity theorem it is also closed, so by connectedness it is all of $X$.

Both hypotheses are needed. On the non-compact manifold $\C^n$, the coordinate function $z_1$ is holomorphic and not constant. On a disjoint union of two compact complex manifolds, take the function equal to $0$ on one piece and $1$ on the other. It is holomorphic and not constant; in general $f$ is only locally constant.

Consequence: a compact connected complex manifold of positive dimension is not a complex submanifold of any $\C^N$, since the coordinate functions restricted to it would be constant. For example, complex projective space does not embed holomorphically in $\C^N$.

Going deeper

For a student

This result is the several-variable version of the fact that a compact Riemann surface has no nonconstant holomorphic functions, and it is a cousin of Liouville's theorem. Because global holomorphic functions are useless on a compact complex manifold, one studies instead meromorphic functions and holomorphic sections of line bundles. The opposite extreme is a Stein manifold, which has enough holomorphic functions to embed in some $\C^N$.