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 good online reference 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:

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    $\begingroup$ Your second question is in the "you can't seriously expect someone to explain that in a MathOverflow post" pile. You should go read a book on the subject, like Ana Cannas da Silva's introduction to symplectic geometry. I wouldn't vote to close the question, but I think it would be justified under the "MO is not an encyclopedia" clause of the FAQ. $\endgroup$
    – Ben Webster
    Nov 7, 2009 at 18:32
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    $\begingroup$ If you want a manageable answer you might want to do come up with a more specific question inside your bigger one. It's a good question, but a proper answer would run to dozens of pages. $\endgroup$
    – Ben Webster
    Nov 7, 2009 at 19:02
  • $\begingroup$ Well, it's not the answer I hoped for. I now changed the post so that the main question is "provide a reference". To clarify: it has to be something I can get on the internet though, thus it can't be really a physical book. $\endgroup$ Nov 7, 2009 at 19:04

2 Answers 2


Ben is right. Consider the example of $\mathbb{C}P^1$, which is an ordinary sphere. The action of $T$ is a rotation around the vertical axis, and the orbits are latitude lines. The action of $V$ pushes the sphere from the south pole to the north pole, and the orbits are meridian lines. The symplectic form is just the area form and the action of $V$ is clearly not area-preserving.

As for your second question, at a technical level Ben is also right. Each of your "whys" is really one part of the Guillemin-Sternberg structure theorem, at least if you take the question for half-dimensional toric actions on a compact symplectic manifold. My impression is that the structure theorem is not all that easy.

But I can think of something else to say. If you take your question for toric varieties in the sense of Fulton's book, then essentially it's all true by construction. You can think of a toric variety as the answer to the question "How can we generalize the projection from the sphere to the interval, keeping all important properties?" If you start with a convex polytope, you should try to build a singular fibration over it such that the fiber over a point on a $k$-face is a $k$-torus. If it is a rational convex polytope that contains the origin, then the definition of a projective toric variety is an organized solution to this question. You build the symplectic structure and the complex algebraic structure so that it all works.

For me at least, it is helpful to first think about the case when the polytope is integrally simple, so that the variety is a manifold. Then the case when it is rationally simple, so that the variety is an orbifold.


I don't think the action of V you indicate is symplectic. It's action on the tangent space at a fixed point is just multiplication by a real scalar on a 2 dimensional space, which doesn't have an invariant symplectic form.


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