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,$b^1 \neq 0$.


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 $M$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.