Are there compact complex manifolds with non-constant pluriclosed functions?

Question

What are some examples of a (connected) compact complex manifold with a non-constant global complex valued function, $f$, such that $\partial {\bar{\partial}} f = 0$.
In other words, what are examples of a connected compact complex manifold with Aeppli cohomology, $H^{0,0}_A > 1$ ? It is clear such a manifold must have non-vanishing first betti number,$b^1 \neq 0$.

Answer 1

A function $f : M \to \mathbb{C}$ satisfying $\partial\bar{\partial}f = 0$ is called pluriharmonic.

In any holomorphic coordinates $(z^1, \dots, z^n)$, a pluriharmonic function satisfies $\partial_{z_i}\partial_{\bar{z}_j}f = 0$ for all $1 \leq i, j \leq n$. In particular,

$$0 = \sum_{j=1}^n\partial_{z_j}\partial_{\bar{z}_j}f = \sum_{j=1}^n\frac{1}{4}(\partial_{x_j}^2f + \partial^2_{x_{n+j}}f) = \frac{1}{4}\Delta f = \frac{1}{4}\Delta u + \frac{i}{4}\Delta v$$

where $z_j = x_j + ix_{n+j}$ and $f = u + iv$. Therefore, in local holomorphic coordinates, the real and imaginary parts of $f$ are harmonic.

Let $K$ be the maximum of $u = \operatorname{Re}f$ and choose $p \in u^{-1}(K)$. If $(U, (z^1, \dots, z^n))$ is a holomorphic coordinate chart with $p \in U$, then $u|_U : U \to \mathbb{R}$ is harmonic and $u(q) \leq u(p)$ for all $p \in U$. By the maximum principle, $u|_U$ is constant with constant value $u(p)$, so $U \subseteq u^{-1}(K)$ and hence $u^{-1}(K)$ is open. On the other hand, $u$ is continuous so $u^{-1}(K)$ is closed. It follows that $u$ is constant on each connected component of $M$.

Applying the same argument to $v = \operatorname{Im} f$, we conclude that $v$, and hence $f$, is constant on each connected component of $M$. Therefore $H^{0,0}_A(M) \cong \mathbb{C}^d$ where $d$ is the number of connected components of $M$.