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Suppose $X$ is a Fano manifold of Picard rank one such that the cotangent bundle $T^*X$ is completely algebraically integrable system. Let $\mu: T^*X \to \mathbb{C}^n$ be the moment map, where $n$ is the dimension of $X$. It is known that a general fiber of $\mu$ is irreducible.

Question: Is it possible to impose some condition on $X$ such that $W=: \{v \in \mathbb{C}^n: \mu^{-1}(v) \text{ is reducible }\}$ has codimension $\ge 2$ ?

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  • $\begingroup$ by reducible do you mean disconnected fibres also? or just reducible and connected? also it might be helpful if you also put some condition on the singularities of the fibres. $\endgroup$
    – S.D.
    Oct 14, 2022 at 17:31
  • $\begingroup$ By reducible, I meant disconnected fiber also. What would be the answer for connected reducible case ? $\endgroup$
    – LAPRAS
    Oct 14, 2022 at 17:36
  • $\begingroup$ We do not have any control on the singularities on the fibers. $\endgroup$
    – LAPRAS
    Oct 14, 2022 at 17:49

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