LECTURE 2. THE GEOMETRY OF THE MOMENT MAP 305We see that the moment polytope encodes some information about the manifoldM; more refined information is encoded in the pushforward function f, which is
supported on μ(M). The function f is polynomial on regions oft* consisting of
regular values of μ, but it or its derivatives may have discontinuities along the
hyperplanes which comprise the singular values of μ. (See [17], Sections 3.3 and
3.5.).
It is possible (see [17], Section 3.3) to write the function fas a linear combina-
tion of functions which encode the pushforward of the Liouville measure under the
moment map for the (linear) action of T on a fibre of the normal bundle to each
component of the fixed point set (which is a symplectic vector space).
2.4. Operations on moment polytopes
2.4.1. Reduction in stages, and slices through moment polytopes
Suppose a compact Lie group G acts in a Hamiltonian fashion on a symplectic
manifold M, and H is a normal subgroup of G. (For example, this is the case if
both Hand Gare tori.) Suppose also that 0 is a regular value for μHand μc. Then
the symplectic quotient μfi^1 (0)/ His acted on naturally by the quotient group G / H ,
and this action is Hamiltonian; furthermore the symplectic quotient of μfi^1 (0)/H
by G/H is naturally isomorphic to μ(-/(0)/G. (This result is known as "reduction
in stages". )
Let M be a symplectic manifold equipped with the Hamiltonian action of a
torus T. Let H C T be a Lie subgroup of T (so H is a torus whose dimension is
smaller than the dimension of T). Let μ r : M ----+ Lie(T) and μH : M ----+ Lie(H)
be the moment maps: recall that μH = 7fH o μ r where 7fH : Lie(T) ----+ Lie(H) is
the standard projection.
For any T/ E Lie(H)* we may form the reduced space M 71 = ¢fi^1 (TJ)/H. This is
equipped with a Hamiltonian action of T / H.
We then see that the moment polytope for the action of T / H on M 71 is given
by a slice through the moment polytope μr ( M), as follows:
2.4.2. Projections of moment polytopes
The previous section described reduction in stages, and explained that a slice
through the moment polytope was the moment polytope for the action of a quotient
group on the reduced space obtained by reducing with respect to the action of a
subgroup. The moment polytope for the action of a subgroup H can similarly be
described in terms of the moment polytope for the action of the original group T : if
M, T and Hare as above, then it follows from Proposition 1.6 (a) that the moment
polytope μH(M) for the action of Hon Mis given by the projection 7rH(μr(M)).
2.5. Symplectic cutting
Let M be a symplectic manifold equipped with the Hamiltonian action of the circle
group 51. We may define the symplectic blowup or E -symplectic cut space M< as
follows. We endow C with the standard symplectic form dx /\ dy; then the moment
map for the action of 51 on C by rotation is μ( z) = -~lzl^2. Hence if the moment