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You may be understandably confused by a terminological inconsistency. People study so-called "Hyper-Kähler manifolds with torsion" (a.k.a. HKT manifolds), which by definition have 3 complex structur …

I think not. Note that replacing $\sigma$ by $\sigma - \omega$ reduces us to the case $\omega=0$. Your question in that case is whether every real $d$-exact 2-form is a $(1,1)$-form. Now unless I'm m …

Compact case (since you mention $\mathbf C^2/\mathbf Z^4$): For $M$ to be Kähler its first Betti number must be even, and conversely every compact complex surface with even $b_1(M)$ is Kähler (Kodai …

No. In $\mathbf C^2$ with standard 2-form and complex structure, the real span of $U=\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)$ and $V=\left(\begin{smallmatrix}i\\1\end{smallmatrix}\right) …

I would recommend the Tralle-Oprea book, Symplectic manifolds with no Kähler structure.

I think you have in mind the integrability (a.k.a. "flatness") problem for $G$-structures. Beyond the cases mentioned at that link (symplectic, Kähler, and complex structures, corresponding to $G=Sp(n …

Flag manifolds have the form $G/C(S)$ where $C(S)$ is the centralizer (in $G$) of its center $S$ (a torus). Most $G/H$ in the list don't have this form, for $H$ has discrete center in each case excep …

The comments make your second question moot unless reformulated, right? For the first, this goes back to Borel (1954, Thms 1 & 2); more details in e.g. Serre (1954, Thms 1,2,3 and remark following Th …

I may be missing something :-) I'm not seeing where Kronheimer shows $$ \text{hyper-Kähler + simply connected} \Rightarrow \text{ALE} \tag{?} $$ as you claim. But in [2] he shows that "every hyper-Käh …

Flag manifolds $G/C(S)$ even exhaust homogeneous symplectic manifolds of $G$: Borel-Weil (1954, Thm 1). Also restated with fewer details in (1954, Thm 1).