Timeline for Why are holomorphic $p$-forms parallel?

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May 25, 2022 at 13:54	comment	added	Ben McKay		The proof surely uses the Bochner technique: comparing two different Laplace-like operators, differing by Ricci, and uses the vanishing of Ricci on the hyperkaehler.
May 25, 2022 at 13:17	history	edited	Nico Berger		edited tags
May 24, 2022 at 14:53	comment	added	Nico Berger		Thank you! In "Calabi-Yau Manifolds and Related Geometries" by Gross, Huybrechts, Joyce, Prop. 23.3, one wants to show that $H^0(X, \Omega_X^k)$ is one-dimensional, if $k$ even, and vanishes otherwise for $X$ hyperkähler. The proof says "Any form in $H^0(X, \Omega_X^2)$ is parallel" and later that for $k>2$ one proceeds analogously. From the holonomy principle one can deduce the assertion about the dimension for parallel sections. I thought one uses the claim I mentioned in my question to deduce the desired properties of $H^0(X, \Omega_X^k)$. Is the claim maybe true for Calabi-Yau manifolds?
May 24, 2022 at 14:39	comment	added	Michael Albanese		As stated, your claim is false. For example, the existence of a non-zero parallel form $\alpha \in H^0(X, \Omega^n_X)$ implies that the holonomy is contained in $SU(n)$, but this is only true for Calabi-Yau manifolds.
May 24, 2022 at 13:57	history	edited	Nico Berger	CC BY-SA 4.0	added 151 characters in body
May 24, 2022 at 13:38	history	asked	Nico Berger	CC BY-SA 4.0