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Let $(M,\omega)$ be a symplectic manifold endowed with a hamiltonian action of a torus $T$. Let $\mu : M \longrightarrow {Lie(T)}^*,$ be a moment map associated to this action. Let $S_M =\bigcap\limits_{m \in M} Stab(m)$, and $s_m$ be its lie algebra.

I have two questions about the moment map:

I) Why is the moment map constant on each connected component of $M^T$ ?

II) I know that for each $m \in M $, the image of tangent map of $\mu $ at m is $ Im(T_m \mu)={(s_m)}^\bot = \lbrace \eta \in Lie(T) \mid \langle \eta , X \rangle = 0 , \forall X \in s_m \rbrace $. How does this imply that the image of M by the moment map is an affine space directed by the linear space ${(s_m)}^\bot$ ?

Maria
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    More information in this book of Jean-Louis Koszul, especially for the case of non-null cohomology based on Jean-Marie Souriau works: https://link.springer.com/book/10.1007%2F978-981-13-3987-5 – Frederic Barbaresco Nov 01 '20 at 07:18

1 Answers1

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I think it is helpful always to build intuition for Hamiltonian circle actions and then bring it over to the general case.

For a Hamiltonian circle actions there is a vector field $X$ (i.e. the integral curves of the vector field are the orbits of $S^1$-action) and smooth function $H : M \rightarrow \mathbb{R}$, subject to the equation: $$dH = \omega(X,\cdot) .$$

Now note that by the above identity a point $p$ is fixed by $S^{1}$ $\iff$ $X|_p=0$ $\iff$ $dH|_p=0$.

Hence the derivative of the Hamiltonian is exactly zero on components of the fixed point set and hence $H$ has to be constant on them. For higher dimensional torus actions picking a basis for the Lie algebra and applying this argument to each of the components give the argument.

For $2$ I think you should clarify since the image of the moment map is not always an affine space although it is a convex polytope (By AGS convexity theorem).

Nick L
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  • Thank you so much for your answer ! But i have a question about the point where you mentioned that ${dH|}_p =0 ,$ for p fixed point by $S^1$, implies that the derivative of the Hamiltonian is exactly zero on components of the fixed point set . Why does not this (${dH|}_p =0 ,$ for p fixed point by $S^1$) imply that the derivative of the hamiltonian is zero on all the fixed point set and hence H is constant on it ? – Maria Oct 31 '20 at 14:07
  • Concerning the second question, I read in a paper that from the equation $d \langle \mu , X \rangle = \omega(X_M , .), \forall X \in Lie(T)$, we conclude that $\mu$ send M to an affine space whose direction is the vector space $(s_M)^\bot $? – Maria Oct 31 '20 at 15:21