# algebraic momentum map

1. Let $$T$$ be a linear algebraic torus over $$\mathbb C$$ and $$X$$ be a smooth quasi-projective symplectic $$T$$-variety. Also, assume that the action of $$T$$ is free and $$X/T$$ exists as a smooth variety. Is this action Hamiltonian?

2. If the above action is Hamiltonian let us denote the algebraic moment map $$X\rightarrow t^{\vee}$$ by $$\mu$$. Does there exist a one-to-one correspondence, via $$\mu$$, between the symplectic leaves of $$X/T$$ and the coadjoint orbits of $$T$$ (which are just points of $$t^{\vee}$$ in this case)?

This is an affirmative answer to $$1$$ (assuming by Hamiltonian you mean in the sense of real symplectic geometry).

Yes. Due to Sumihiro's theorem (+ standard facts about Hamiltonians). In fact the assumptions smooth quotient and symplectic are not necessary, just smooth and quasi-projective is enough. Throughtout when I mean symplectic I mean in the real sense.

Sumihiro's Theorem ("Equivariant completion", J. Math. Kyoto Univ., 14: 1–28) implies that: For any normal, quasi projective variety with an algebraic $$T$$-action there exists $$N>0$$ and an $$T$$-equivariant embedding $$\phi : X \rightarrow \mathbb{CP}^N,$$ for some algebraic $$T$$-action on $$\mathbb{CP}^N$$.

Next, it is known that every algebraic torus action on projective space is linear, so in a particular homogeneous coordinate system of $$\mathbb{CP}^N$$ the action is given by $$z.[z_{0}: \ldots : z_{n}] = [z^{a_{0}}z_0 : \ldots : z^{a_{n}}z_n] ,$$ for some integers $$a_{0}, \ldots a_n$$.

Then, this action is Hamiltonian (with respect to the Fubini-Study Kahler form on $$\mathbb{CP}^n$$) with Hamiltonian $$H([z_{0}: \ldots : z_{n}] ) = \frac{\sum a_i |z_i|^2}{\sum|z_i|^2}.$$

Finally, there is a general fact about Hamiltonian torus actions, the proof is linear algebra.

Fact: Let $$(M,\omega)$$ be a symplectic manifold with a Hamiltonian $$T$$-action and moment map $$F$$. Then if $$N \subset M$$ is an invariant, symplectic submanifold then the restricted action on $$N$$ is Hamiltonian with moment map $$F|_{N}$$.

So applying this result to the symplectic submanifold $$\phi(X)$$ to obtain that the action is Hamiltonian. I could probably find references for these statements if you'd like.