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I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 69 is this exercise:

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Can some one give me a hint? I'm kinda new to the subject.

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  • $\begingroup$ I'm rusty on this kind of thing, but I would start with the case $M = \mathbb R^{2n}$. I'm pretty sure this is ultimately just a statement about eigenspaces. Once you have the underlying linear algebraic fact identified, globalizing should be pretty straightforward. $\endgroup$ – Tabes Bridges Oct 11 '18 at 17:35
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    $\begingroup$ Just a small comment: Holomorphic $1$-forms on a complex manifold are a very special sort of $(1,0)$ form, so your title is a bit off. (And you really need an integrable almost complex structure for it to make sense.) $\endgroup$ – Ted Shifrin Oct 11 '18 at 18:08
  • $\begingroup$ @TedShifrin you are right! $\endgroup$ – Hurjui Ionut Oct 12 '18 at 12:09

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