an interior fixed point has one positive weight and one negative
weight, a maximal fixed point has both weights non-positive, a
minimal fixed point has both weights non-negative;
a fixed point has a weight 0 if and only if it lies on a fixed surface.
For example, in Figure 2, the isotropy weights corresponding to the
left interior vertex are {—2,1}, and those corresponding to the top
vertex are {—2, —1}.
Note that the graph and its integer labels depend only on the man-
ifold and circle action. The real labels are essentially determined by
the cohomology class of the symplectic form:
Lemma 2.5. The cohomology class of UJ determines the moment map
values at the fixed points up to a simultaneous shift of all these values
by the same amount, and it determines the normalized symplectic areas
of fixed surfaces.
Proof. The second part is clear. For the first part, let p and q be fixed
points with $(p) $(q). Choose any smooth path, j(i), 0 t 1,
from p to q. Denote := [0, 2TT] X [0,1], and define / : M by
f(s,t) = e*-
(*). Then juf*uj = 2n f^
* W U M = M$(P) - % ) ) .
Since / defines a cycle in homology, this integral depends only on the
cohomology class of u, hence so does the difference $(p) $(q). D
In particular, if p and q are the north and south poles of a Z^-sphere,
the difference $(p) $(g) is equal to the symplectic area of the sphere
times k/27r.
2.2. Kahler toric varieties. Take a compact symplectic manifold
(M, UJ) of dimension 2n with a Hamiltonian action of an n-torus,

x ... x 5
, and a moment map $ : M
meaning a map whose
n coordinates generate, via (1.1), the actions of the n circles. Such a
triple (M, u,$) is called a Delzant space.
By the convexity theorem [GS1, A], the image of the moment map
is a convex polytope. By Delzant's theorem [De], this polytope deter-
mines the Hamiltonian space "up to equivariant symplectomorphism,
and the space is a Kahler toric variety, meaning that it admits a com-
plex structure such that the torus T acts holomorphically and the sym-
plectic form UJ is Kahler. See [De] and [G]. The polytopes that arise in
this way are called Delzant polytopes. When n = 2, these are exactly
those polygons in R2 that have the following properties:
the slopes of the edges are rational or infinite;
every two consecutive edges have integral outward
normal vectors (fc, b) and (£;',
bk' = 1. (2.6)
Previous Page Next Page