Consider a holomorphic function $u:C\to C^n$, as $|u|^2 $is sub-harmonic, it satifies a maximum principle.
Do some general kind of complex manifold enjoy such property? Say, square of some distance function(or similar function ) will be sub-harmonic?
If $f_1$, $f_2$, ..., $f_n$ are holomorhic functions on an open set $U$ in $\mathbb{C}^k$, then $\sum_{i=1}^n |f_i|^2$ has no local maximum. Since complex manifolds are locally $\mathbb{C}^k$, this is also true if $U$ is an open set in a complex $k$-fold.
In fact more is true: The signature of the Hessian of $\sum_{i=1}^n |f_i|^2$, as a quadratic form on $\mathbb{R}^{2k}$, has at most $k$ negative signs. The proof is pretty straightforward: Suppose $V \subset \mathbb{R}^{2k}$ is a $d$-dimensional real vector space on which the Hessian is negative definite; we must show $d \leq k$. If not, then $V \cap i V$ is nonzero. Note that $V \cap i V$ is a complex subspace of $\mathbb{C}^k$. We can restrict the $f_j$ to holomorphic functions on a complex line in this vector space and violate the standard maximum modulus principle.
This comes up in the proof that $k$-dimensional Stein manifolds have the homotopy type of a $(\leq k)$-dimensional CW-complex: For such a Stein manifold $X$, let $(f_1, \ldots, f_n)$ be a homolomorphic embedding $X \to \mathbb{C}^n$. After possibly replacing $f_i$ by $f_i-a_i$ for some generic $(a_1, \ldots, a_n) \in \mathbb{C}^n$, the function $\sum |f_i|^2$ is a morse function, and we have just computed that the signatures of the critical points are $+^{2k-d} -^d$ for $d \leq k$.
