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Question about the Kähler structure on generic coadjoint orbits

For the affine case for coadjoint orbits of Souriau, see Jean-Louis Koszul book "Introduction to symplectic geometry" in chapter 4 and 5 where it is explained the Souriau cocycle in cas of ...
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Question about the Kähler structure on generic coadjoint orbits

Put $T = G_r$. We may, and do, assume that $\beta = r$, and simply describe a $T$-invariant complex structure on $\operatorname T_{\mathcal O_r}(r)$. Instead of having one 1-dimensional subspace of $\...
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Can deformation equivalent Kähler manifolds always be obtained by a deformation where all the fibers are Kähler?

I don't think this is known. For hyperkahler manifolds, conjecturally, all smooth complex deformations are class C and birational to hyperkahler. If this is true, your conjecture would follow ...

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