Let $X$ be a Fano manifold (different from $\mathbb{P}^n$) of dimension $n$ and Picard rank one. If its cotangent bundle $T^*X$ admits $n$ independent Poisson commuting regular functions then $T^*X$ is a completely integrable system with the moment map $T^*X \to \mathbb{C}^n$ defined by these functions. For example, if $X$ is the moduli space of vector bundles on curves, then $T^*X$ is completely integrable system.

Question: Is there any other example of such $X$, different from the moduli space of vector bundles ?

  • 2
    $\begingroup$ No. I want $X$ such that its cotangent bundle is integrable system. $\endgroup$
    – LAPRAS
    Nov 26, 2022 at 5:21
  • $\begingroup$ Right, I misread your question. $\endgroup$
    – Sebastian
    Nov 26, 2022 at 6:05
  • $\begingroup$ The homogeneous spaces $G/P$ are also examples of such Fano. I am looking for an example which is different from these two examples. $\endgroup$
    – LAPRAS
    Nov 26, 2022 at 13:24
  • $\begingroup$ How about specializations of $G/P$ as in the article of Pasquier-Perrin? $\endgroup$ Nov 26, 2022 at 13:40
  • $\begingroup$ In a post in mathoverflow.net/questions/210100/… , there is an example where the specialization of a projective homogeneous space is not homogeneous. Does the example works for my question also? i.e., its cotangent bundle is integrable system ? $\endgroup$
    – LAPRAS
    Nov 26, 2022 at 16:40


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