Consider a **toric variety**, defined as a (normal?) complex projective variety $X$ together with an algebraic action of $(\mathbb C^*)^n$ with finitely many orbits. Now we have two "real symplectic" actions there, the one by a real torus $T$, and the one by a real vector space $V$. From the real symplectic geometry I know that such an action always comes infinitesimally from a family of Hamiltonians $\{H_i\}$.

An example would be action of $\mathbb C^*$ on $\mathbb C\mathbb P^1$ and the Hamiltonian should be a suitable function of $\mathrm{abs}\\,(z)$ (its square?) when $z$ is represented as the point on the invariant $\mathbb C$-plane; $S^1$-action generated by this Hamiltonian is rotation around 0.

`(1)`

Why can the action of $T$ be integrated to a global family $\{H^T_i\}$ and not the action of $V$ generally?

The map $x\mapsto \{H_i^T(x)\}$ is called a moment map and I need to learn about its properties: why "polytopes with rational vertices = projective toric varieties" et cetera. I have only book references, and I strongly prefer something downloadable.

`(2)`

What would be a goodonlinereference to learn about this?

Okay, I've **changed the question to community wiki** (since the "provide a good reference" question is expected to have many answers). I'll add the references I found in the list below. If you fell like it, you can put them here as well:

- Original (?) article
*Riemann-Roch for toric orbifolds*by Victor Guillemin - Draft of a book
*Toric Varieties*by Cox et al.