6

YAEL KARSHON

FIGURE

2. Graph for Examples 2.4 and 2.8

the corresponding fixed point set. Additionally, to a vertex that corre-

sponds to a symplectic surface, B, we attach two additional labels: the

genus of that surface, and its normalized symplectic area, defined to be

We call the labels of the vertices "moment map labels", "area labels",

and "genus labels", respectively.

Remark 2.3. In our figures, we will often omit some of the labels. The

moment map labels will be indicated by the height of a vertex in the

plane. Those vertices that correspond to fixed surfaces will be drawn

fatter than those that correspond to isolated fixed points.

Example 2.4. Let M be the product of two spheres of radius 1, each

with the standard area form. Let the circle act by rotating the second

sphere at twice the speed of the first: A • (u,v) = (\u,

\2v),

where

(u, v) G

S2

x

S2

C

R3

x

R3,

and where the action on

S2

is by rotations

in the first two coordinates of

R3.

There are four fixed points: (n,ri),

(s,n), (n, s), and (s, s), where n and s are the north and south poles

of S2. There are two Z2-spheres: {n} x S2 and {s} x S2. The moment

map is J(t?, v) = Us + 2v3. The graph is shown in Figure 2.

Lemmas 2.1 and 2.2 force the graph to have a simple shape:

• there is a unique top vertex and a unique bottom vertex;

• the edges occur in a finite number of branches, with the moment

map labels increasing along each branch; a branch needn't reach

an extremal vertex;

• an extremal vertex is reached by at most two edges; an extremal

"fat" vertex is not reached by any edge.

The isotropy weights at the fixed points can be read from the graph:

• for k 2, a fixed point has an isotropy weight —A : if and only if

it is the north pole of a Z^-sphere, and it has an isotropy weight

k if and only if it is the south pole of a Z^-sphere;