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Yes. I assume that your $M_n$ is what is more usually denoted $\overline{M_{0,n}}$. Then the answer is yes, there is a natural map $\overline{M_{0,n}} \twoheadrightarrow M_L$, for each $L$. Specifica …

The basic answer is "yes, of course, because the toric variety is uniquely determined by the polytope. But no, because it's the wrong polytope for the question of Fanoness." The question is whether t …

If $G$ acts on $M$ (both compact and finite-dimensional) preserving the symplectic form, and $M$ is simply-connected, the action is Hamiltonian. Then $M$ maps symplectomorphically to a coadjoint orbit …

I assume you mean that $T$ acts preserving each fiber. Then the flatness says that the multigraded Hilbert polynomial is constant. As the Duistermaat-Heckman measure is the leading-order behavior of t …

Not a lot. Instead you should generalize $Vol(M)$, the pushforward of Liouville measure to a point, to $Vol_G(M)$, the pushforward along the moment map. I'll assume $G=T$ for convenience. The result i …

The faces are all of the following form: $w W_P / Stab_W(\xi)$, where $W_P$ varies over the subgroups generated by subsets of the simple reflections. In particular, for $\xi$ regular, the number of th …

Certainly, if $G$ has a central extension, $G$ will act on the coadjoint orbits of the central extension. So I think your question might be "When can we be sure that our group has no central extension …

Here's two natural things to ask about any compact group action on a compact manifold: (1) what are the (finitely many) conjugacy classes of stabilizer groups? Assume our group is a torus, so we can o …

If $M \to X$ is smooth and proper, and $M$ is K\"ahler, then the fibers are all symplectomorphic. (Proof: the Levi-Civita connection generates symplectomorphisms.) The family of elliptic curves was al …

Hello again. Yes, it's true. The more general statement you want is, let $X$ be projective with a $K$-equivariant ample line bundle ${\mathcal O}(1)$. For each $n$, let $\mu_n$ be a measure on ${\ma …

That particular example is about branching from $SO(5)$ to $SO(4)$. In general, the branching rule from $SO(n)$ to $SO(n-1)$ is multiplicity-free and well-known; it's in, e.g., Zhelobenko's book. So t …

Since you've already gotten lots of classical mechanical answers, I'll give my favorite source of Lagrangians. Let $\sigma$ be a holomorphic section of a Hermitian line bundle $\mathcal L$ with curvat …

Let $F\to E\to B$ be a smooth proper fibration over a connected manifold $B$, where the total space is symplectic and the fibers are smooth and symplectic. Then one can use Moser's theorem to generat …

It depends on whether you think of the symplectic form as giving you a map $TX \to T^*X$ or vice versa. When you restrict to $\mu^{-1}(\chi)$, you only have a "presymplectic form", which gets you a ma …

I believe the following way (Kostant's, 1970) to be the best way to think about the Hamiltonian condition. First, "why" is there a central extension $H^0(M; {\mathbb R}) \to C^\infty (M) \to symp(M)$ …