PERIODIC HAMILTONIAN FLOWS ON FOUR MANIFOLDS 5
p
\ """"""/ k times
FIGURE
1. A ^-sphere
important examples: Delzant spaces. In section 2.3 we discuss the
Duistermaat-Heckman measure in relation to the graph.
2.1. The graph. Let (M,u) be a compact symplectic four-manifold
with a Hamiltonian circle action and a moment map $ : M —• M. In
Appendix A we recall the following facts:
Lemma 2.1. Each component of the fixed point set is either a single
point or a symplectic surface. The maximum and minimum of the
moment map is each attained on exactly one component of the fixed
point set. Fixed points on which the moment map is not extremal are
isolated.
We call a fixed point extremal (maximal or minimal) if it is an
extremum for the moment map; otherwise, we call it an interior fixed
point.
Lemma 2.2. For each integer k 2, consider the set of points whose
stabilizer is equal to the cyclic subgroup of S1 of order k,
Zk = {A e S1 | Xk = 1}.
Each connected component of the closure of this set is a closed sym-
plectic two-sphere, on which the quotient circle, S1 jZ\~, acts with two
fixed points.
We call such a sphere a Zk-sphere.
We now construct the graph associated to (M, a;, $):2
To every component of the fixed point set we assign a vertex, and
to every Zk-sphere we assign an edge connecting the corresponding ver-
tices. We label each edge by the isotropy weight, k, of the corresponding
Zk-sphere. We label each vertex by the value of the moment map on
2Earlier
versions of this manuscript contained an equivalent, but slightly more
complicated, construction.
Previous Page Next Page