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.