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.

  • 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. – Tabes Bridges Oct 11 at 17:35
  • 1
    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.) – Ted Shifrin Oct 11 at 18:08
  • @TedShifrin you are right! – Hurjui Ionut Oct 12 at 12:09

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