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Consider the hamiltonian action of $T^2$ on $\mathbb{CP}^2$ :

$$ \varphi: ((e^{i\theta_1},e^{i\theta_2}),[z_0:z_1:z_2]) \longmapsto [z_0:e^{i\theta_1}z_1,e^{i\theta_2}z_2].$$

I've read that its moment map is

$$ \mu (z_0,z_1,z_2) = -\tfrac{1}{2} (\tfrac{|z_1|^2}{|z_0|^2+|z_1|^2+|z_2|^2},\tfrac{|z_2|^2}{|z_0|^2+|z_1|^2+|z_2|^2}).$$

How can I show this with a explicit calculation of $d\mu^X$ and $i_{X^\#}\omega $ to show they are equal? I could do this with the same action but on $\mathbb{C}^2$, with polar coordinates. Would this work in this case, if yes what would be the equivalent of polar coordinates for projective space?

Reb
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1 Answers1

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The action on $\mathbb{C}^3$ given by $(e^{i\theta_1}, e^{i\theta_2}) \cdot (z_0, z_1, z_2) = (z_0, e^{i\theta_1}z_1, e^{i\theta_2}z_2)$ has for its moment map $\mu(z_0, z_1, z_2) = \frac{1}{2} (|z_1|^2, |z_2|^2)$ (up to sign which depends on convention).

This action preserves the unit sphere $\lbrack |z_0|^2 + |z_1|^2 + |z_2|^2 = 1 \rbrack$ and commutes with the Hopf fibration $\pi : S^5 \to \mathbb{C}P^2 : (z_0, z_1, z_2) \mapsto [z_0 : z_1 : z_2]$, so it induces an action on $\mathbb{C}P^2$ which is precisely the one you are considering.

You can restrict the above map $\mu$ to $S^5$ and observe that it coincides (on $S^5$) with the map $\mu'(z_0, z_1, z_2) = \frac{1}{2} \, \left( \frac{|z_1|^2}{|z_0|^2 + |z_1|^2 + |z_2|^2} \, , \, \frac{|z_1|^2}{|z_0|^2 + |z_1|^2 + |z_2|^2} \right)$. Observe that $\mu'$ is constant on the fibers of the Hopf fibration ; In fact, it is invariant under the action of $\mathbb{C}^{\times}$ on $\mathbb{C}^3$ which yields $\mathbb{C}P^2$ as a quotient. Therefore, $\mu'$ determines a well-defined map $M$ on $\mathbb{C}P^2$.

In order to prove that $M$ is the moment map for the induced action on the projective space, the best is to lift everything on the sphere. Observe that $\mu' = \pi^{\ast}M$, that the 'symplectic' form on $S^5$ (induced by its inclusion in $\mathbb{C}^3$) is the pullback by $\pi$ of the Fubini-Study form on $\mathbb{C}P^2$, that $\mu'$ is the 'moment map' for the 'Hamiltonian' action of $\mathbb{T}^2$ on $S^5$ and that the Hamiltonian vector field $\xi^{\sharp}$ on $\mathbb{C}P^2$ associated to any element $\xi = (\xi_1, \xi_2) \in (Lie(\mathbb{T}^2))^{\ast}$ is $\pi$-related to the 'Hamiltonian' vector field $\xi^{\sharp \, '}$ on $S^5$ associated to the same $\xi$. Therefore, the fact that $\mu'$ is a 'moment map' for the 'Hamiltonian' torus action on the sphere (i.e. $d\mu'(\xi) = \iota_{\xi^{\sharp \, '}} \, \omega$) implies that $M$ is the moment map of the Hamiltonian torus action on $\mathbb{C}P^2$ (i.e. $dM(\xi) = \iota_{\xi^{\sharp}} \, \omega_{FS}$).

Jordan Payette
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  • Ok, thank you very much! So the idea is checking that all the elements: action, moment map, fundamental vector field are well defined when doing quotient from $S^3$ to $\mathbb{CP}^n$, no? So it induces to new hamiltonian action and moment map in projective space. I will have to read about Fubini-study form though, as I didn't know about it. – Reb Apr 13 '17 at 13:34
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    When you say $S^3$, do you mean $S^5$? – Amitai Yuval Apr 13 '17 at 13:45
  • @AmitaiYuval Absolutely, thank you for pointing out this 'typo'. I edited the answer to fix this mistake. – Jordan Payette Apr 13 '17 at 14:57
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    @Rob You should read about the FS symplectic form on projective space; After all, if you are to speak of an Hamiltonian action on this space, you better have at hand a specific symplectic form! The FS symplectic form is defined by symplectic reduction. The sphere $S^5 \subset \mathbb{C}^3$ is a coisotropic manifold for the usual ambient symplectic structure: As such it has a canonical foliation by 1-manifolds and the quotient of $S^5$ by this foliation inherits a symplectic structure. Here, the foliation happens to be the Hopf fibration and the quotient is $\mathbb{C}P^2$, with the $FS$ form. – Jordan Payette Apr 13 '17 at 15:04
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    Because of this particular construction of the projective space (and of its symplectic form), one can argue that everything which happens on the projective space is the 'shadow' of something which happens on the sphere, and hence on the ambient complex space by extending in an appropriate way. Reciprocally, everything happening in the ambient space which behaves 'well' with respect to the quotient map to the projective space can be essentially recovered by the preceding process. What I say here is admittedly informal: It is more of a 'guiding principle' when playing with projective spaces. – Jordan Payette Apr 13 '17 at 15:12