326 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS
are the vector fields X on T*JR.^3 generated by the action of X E JR.^3 = g on
T*JR.^3 (where G = S0(3) acts on JR.^3 by rotations).
5. (a) Show that if a submanifold N of a symplectic manifold Mis preserved
by the action of an almost complex structure J on M which is compatible
with the symplectic form (in other words if N is an almost complex sub-
manifold of M with respect to J) then the symplectic form restricts to a
nondegenerate form on N. (An almost complex structure J is compatible
with the symplectic form w if(i) w(JX, JY) = w(X, Y)
(ii) w(JX,X) > 0
for all vector fields X and Y on M.)
(b) Assume that I : en __, e is a holomorphic function. Show that if 0 is aregular value of I then 1-^1 (0) is a symplectic submanifold of en (equipped
with its standard symplectic structure).
( c) Assume that I : en __, e is a homogeneous polynomial function (in otherwords I (>-.z) = Ad I (z) \;/).. E C*). Show that if 0 is a regular value of I then
{[z1: ···:Zn] E epn-l: (z1, ... ,zn) E 1-^1 (0)} is a symplectic submanifold
of epn-l (equipped with its standard symplectic structure).
- (Orbits of Hamiltonian group actions are isotropic) Let M be equipped with
the Hamiltonian action of a compact Lie group G. Show that the orbits of the
action of G on μ -^1 (0) are isotropic with respect to the symplectic structure. - (Symplectic slices) Le:t Y be a symplectic manifold equipped with the Hamil-
tonian action of a torus T which is the maximal torus of a compact Lie group
G with moment map μ r : Y --> t *.
Define
M ~f Y xr G = {(y,g) E Y x G: (y,g) '.:::'. (ty, tg) fort ET}.
Define a symplectic structure on M on with respect to which the action of
G is Hamiltonian. Exhibit a moment map μ a : M --> g* for the action of Gon M. What is μ 01 (t)?
- Show explicitly that the diagonal elements of matrices conjugate (under
SU(2)) to diag(27ri, -27ri) in su(2) are of the form Bdiag(27ri, -27ri) where
BE [-1, l]. - Describe the symplectic quotient Mm 1 , ... ,mn obtained from the action of
U(l) on en specified by (nonzero) integers (m 1 , ... , mn)- If Mm 1 ,.. .,mn is
a smooth manifold, show that it is equipped with the action of an (n-1)-
dimensional torus and describe the moment polytope for this torus action.- Show that the symplectic volume of a toric manifold is equal to the Euclidean
volume of its moment polytope.
11. Show that eP^2 is the toric manifold whose moment polytope is the isosceles
right triangle.