EXERCISES 327
12. Exhibit explicitly the subsets of CP^3 for which the stabilizer under the
standard action of U(1)^3 is
(a) isomorphic to U(l)
(b) isomorphic to U(1)^2
( c) isomorphic to U ( 1 )^3.
Describe the images of these subsets under the moment map μU(l)3: show
that they are
(a) facets (i.e. 2-dimensional faces)
(b) edges (i.e. 1-dimensional faces)
( c) vertices.
- (Example of projection of moment polytopes) (a) Give an approxi-
mate sketch of the image μH(M) in the case when M = CP^3 , T = U(1)^3 c
U(1)^4 acting in the usual way (so the moment polytope is a 3-simplex in
ffi.^4 ) and H = U(1)^2 acting via some embedding in U(1)^3 : choose the ortho-
complement of Lie(H) in Lie(T) to be some axis !Rv through the origin in
ffi.^3. (The image will depend on the axis !Rv chosen.)
(b) What are the regular values for the moment map of H?
- Suppose (['.P+q is acted on by U(l) by the map
u E U(l): (z 1 , ... , Zp, W1, ... , wq) ~ (u z 1 , ... , u zp, u -^1 w 1 , ... , u-^1 wq)·
How does the reduced space μ -^1 (E)/U(l) change as we pass from E > 0 to
E < O?
(Consider in particular the case p = 1.)
(This problem is the subject of the paper [20]: see [5], Section 5.4 for a
discussion.)
15. (Pushforward of the moment map under a collection of weights)
Let T = {(z 1 ,z 2 ,z3) E U(1)^3 : z 1 z2z3 = 1} act on C^3 via the weights
/3 1 , /32, (33 given by
/31(X1,X2,X3) = X1 - X2,
/32(X1, X2, X3) = X2 - X 3,
f33(X1,X2,X3) = X1 - X3.
(Thus (3 3 = (3 1 + /3 2 and /3 1 and /3 2 form an angle of 2?T /3 in Lie(T) ~ ffi.^2.
The f3J are the positive roots of SU(3).)
Compute the pushforward of the Liouville measure under the moment map
for this action of T, or equivalently the pushforward of Lebesgue measure
from (ffi.+)^3 to Lie(T) under the map
(y1,Y2,y3) ~ LYJ/3j·
j
(You will recover a piecewise linear function which is supported on the cone
C(/3 1 ,/32,/3 3 ) and is equal to 0 on the boundary of this cone.)
