8 YAEL KARSHON
FIGURE
3. Delzant polygons for Example 2.8
The pre-image in the manifold of a vertex of the polygon is a fixed
point for the torus action. The pre-image of an edge with slope k/b
is an invariant 2-sphere whose stabilizer is the subgroup of S1 x S*1
consisting of the elements of the form (Afc, A-6). The pre-image of the
interior of the polygon consists of free torus orbits. These facts follow
from the local normal form for Hamiltonian torus actions and from the
connectedness of the level sets of the moment map, and are explained
in Delzant's paper [De].
If we now restrict the action to the sub-circle {e} x 5
1
, we get a
compact four dimensional Hamiltonian S^-space. The moment map
for the Sfl-action is the T-moment map composed with the projec-
tion R2 M to the second coordinate. The fixed surfaces are the
pre-images, under the T-moment map, of the horizontal edges of the
Delzant polygon. Such a surface has genus zero, and its normalized
symplectic area is equal to the length of the corresponding horizontal
edge. The isolated fixed points are the pre-images of those vertices of
the polygon that do not lie on horizontal edges. The Z^-spheres are
the pre-images of edges with slope ±k/b in reduced form, where b is
relatively prime to k. With this information, it is easy to construct the
graph for the
S1
space out of the Delzant polygon.
Remark 2.7. In our figures of Delzant polygons, the dots mark the
weight lattice
Z2
in
R2.
Example 2.8. The Delzant spaces whose polygons are drawn in figure
3 give Hamiltonian
51-spaces
with the same graph, which is the graph
drawn in Figure 2. These
S1
-spaces are all isomorphic to the one
described in Example 2.4. The three polygons all correspond to S2 x S2,
with its standard torus action, composed with an automorphism of the
torus that preserves the second sub-circle.
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