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I am trying to prove the following:

If $f : A \subset M \to \mathbb{R}$ is a $C^{\infty}(M)$ function such that $dd^c f = 0$ then $f = c,$ where $M$ is a compact Kähler manifold, $c\in \mathbb{R}$. We remember that $dd^c = 2i\partial\overline{\partial}$ and $\partial$, $\overline{\partial}$ are de Dolbeault operators.

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    What's $M$ in $C^\infty(M)$? – 2017-01-23
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    Does "cte" stand for "constant"? If so, then this implication is false, even in the case of one variable. The equation $dd^cf=0$ is satisfied by any function that is the sum of a holomorphic function plus an anti-holomorphic function. – 2017-01-23
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    @JackLee, I am sorry for the mistakes. What about now? – 2017-01-23

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Hints:

  1. Every solution to $dd^c f=0$ is harmonic.
  2. Every harmonic function on a compact manifold is constant.
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    For the first assumption, we may use some Kahler identities and the second is immediate, from the short I know, via Hodge theorem. – 2017-01-24