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;
Previous Page Next Page