I am trying to understand paper by Siu "Every K3 surface is Kahler". Let $M$ be a K3 surface.
Siu wrote $H^1 ( M , \mathscr{O}_M ) =0$ without any references. It is written on fifth page. Maybe I missed something, but it seems unlikely.
Question Why $H^1 ( M , \mathscr{O}_M ) =0$ ?
Comment 1 Siu defines K3 surface as "simply connected compact manifold of complex dimension two whose canonical line bundle is trivial".
Comment 2 Why is not it obvious? Assertion would be quite clear if we already knew that $M$ is Kahler. What we actually need to know is that Frölicher spectral sequence degenerates. Indeed, "simply connected" implies $H^1( M , \mathbb{C} ) = 0$.
I want to emphasise, that the purpose of this paper is to prove, that K3 surfaces are Kahler. Therefore, we must not use assumption that $M$ is Kahler.