Let $X$ be a projective manifold and we have a degeneration of fibers such that they are biholomorphic to coadjoint orbits, i.e, $X\to \Delta$ , and fibers $X_t$ are biholomorphic to coadjoint orbits then does $X_0$ biholomorphic to coadjoint orbit?
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3$\begingroup$ Your question related to Mok rigidity theory. You must impose some sort of positivity on central fiber. For example if $X_0$ is Kahler then we can say something more. Let $S$ be an irreducible Hermitian symmetric space of compact type. Let $π:X→Δ $ be a regular family of compact complex manifolds over the unit disk $Δ$. Suppose $X_t:=π^{-1}(t)$ is biholomorphic to $S$ for $t≠0$ and the central fiber $X_0$ is Kähler. Then $X_0$ is also biholomorphic to $S$ , see arxiv.org/abs/math/9604227 $\endgroup$– user21574Commented Dec 9, 2017 at 8:04
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1$\begingroup$ In general, I think Let $π:X→Δ$ be a regular family of compact complex manifolds over the unit disk $Δ$. Suppose $X_t:=π^{-1}(t)$ is biholomorphic to $S$ for $t≠0$ which $S$ admits Kahler-Einstein metric and the central fiber $X_0$ is Kähler. Then $X_0$ is also biholomorphic to $S$, (by this statement is less related to your question) , note that being central fiber Kahler is strong condition, since in general $X_0$ is Moishezon when general fibers are projective $\endgroup$– user21574Commented Dec 9, 2017 at 8:15
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By "coadjoint orbits" I assume you mean "of a compact Lie group, but then endowed with invariant complex structures". In which case the answer is no. There is a flat family whose general fiber is $\mathbb P^1 \times \mathbb P^1$, a coadjoint orbit of $SU(2)^2$, but whose special fiber is the second Hirzebruch surface (which contains an exceptional curve and so is not a homogeneous space).
answered Jan 4, 2018 at 21:16