are top holomorphic forms exterior products of holomorphic forms on almost complex manifolds?

Question

Let $M$ be a manifold with a non-integrable almost complex structure, and let the form $\omega \in \Lambda^{n,0} TM$ be a holomorphic form, i.e. $\bar\partial \omega = 0$. Is it true then that there exist holomorphic forms $\alpha_1, \ldots, \alpha_n$ such that $\alpha_1 \wedge \ldots \wedge \alpha_n = \omega$?

Answer 1

No. For example, the Fermat quartic surface $X_0^4+X_1^4+X_2^4+X_3^4=0$ in $\mathbb{CP}^3$ is a K3 (so has a nonvanishing holomorphic 2-form) but there are no nontrivial holomorphic 1-forms.