306 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS
map for the action of S^1 on M is denoted μ 1 , the moment map for the diagonal
action of S^1 on M x C is
1 2μ(p,z)=μ1(p)-2lz l ·
The symplectic quotient M, = μ -^1 ( E) I S^1 may be identified as
M, = {p EM: μ1(p) 2: E}/"'
where the equivalence relation "' is defined by
Pl"' P2 <-t μ1(P1) = μ1(P2) = E and P1 = P2S for some s E U(l).
This space is called the E-symplectic blowup or E-symplectic cut space of M: its
construction is due to Lerman [31]. The definition of the symplectic cut space
requires only that the S^1 -action and the moment map μ 1 be defined on an open
neighbourhood U of a closed submanifold N for which N = μ 1 -^1 (E).
If M is equipped with the Hamiltonian action of a torus T which commutes
with the action of S^1 , then so is the cut space M<> and the moment polytope for
M, may be obtained by "cutting" the moment polytope of M (see [15], Chapter 1).
2.6. Torie manifolds
Definition 2.8. A toric manifold is a compact symplectic manifold M of dimension
2n equipped with the effective Hamiltonian action of a torus T of dimension n.Example 2.9. Complex projective space cpn with the obvious Hamiltonian ac-
tion of U(lr c U(l)n+l is a toric manifold.
Example 2.10. Special case: the two-sphere S^2 (with the action of U(l) given by
rotation around one axis) is a toric manifold.2.6.1. Elementary properties of toric manifoldsl. If M is a toric manifold, the fibre of the moment map for the action of T is an
orbit of the action. Hence the symplectic quotient Mt at any value t E t* is a point
(if it is nonempty).- The regular values ofμ are the interior points of the moment polytope P. All
points in the preimage μ-^1 (8P) are fixed points of some one parameter subgroup
of T. Points in the interior of a face P§" of dimension j are fixed by a subtorus
of T of dimension n - j. Hence each fibre of μ over a point in P§" is a quotient
torus of dimension j. In particular the vertices of the polytope are the images of
the components of the fixed point set of the whole torus T , and the inverse image
of a vertex is contained in the fixed point set of T.
- The pushforward function μ * (wn /n!) under the moment map (cf. Section 2.3)
is just the characteristic function of the moment polytope.
2.6.2. Delzant's theoremIn fact toric manifolds are characterized by their moment polytopes. A theorem of
Delzant [12] says that any polytope P satisfying appropriate hypotheses (a simple
polytope) is the moment polytope for some toric manifold; furthermore, if two toric
manifolds acted on effectively by a torus T have the same moment polytope then
they are T-equivariantly symplectomorphic. I shall not comment on the proof of the
second statement. The first statement is proved by constructing a toric manifold