Compactness forces a maximum modulus, and connectedness propagates the resulting local constancy globally.Suggest a slogan
Let \(X\) be a compact connected complex manifold. Then every holomorphic function \(f\colon X\to\CC\) is constant.
Since \(X\) is compact, \(\abs f\) attains its maximum \(M\) at some point. The set \(S\coloneqq\ts{x\in X\st f(x)=c}\), for \(c\) a value with \(\abs c=M\) attained by \(f\), is nonempty and closed. At a point of \(S\), \(f\) read in a holomorphic chart is holomorphic on a connected open subset of \(\CC^n\) with an interior maximum of \(\abs f\); by the maximum modulus principle, applied on complex lines through the point, \(f\) is constant near that point, so \(S\) is open. As \(X\) is connected, \(S=X\).