1. INTRODUCTION

Take a compact symplectic manifold (M, a;) and a smooth function

$ : M —• EL The corresponding Hamiltonian flow is generated by the

vector field £M that satisfies

d& = -i{£M)u). (1.1)

We assume that this flow is periodic, with period 2TT, i.e., that ^M

generates a circle action on M. The function $ is called the moment

map for this action. The triple (M, a;, $) is called a Hamiltonian S1-

space, or a Hamiltonian circle action, or a periodic Hamiltonian flow.

An isomorphism between two such spaces is a diffeomorphism F :

Mi —• M2 such that

F*LJ2

= wi and F*E2 = $1; it follows that F is

S^-equivariant. We always assume that M is connected and that the

circle action is effective.

Hamiltonian actions of other Lie groups are defined in a similar way.

For a group G we get a moment map $ : M — 9* where 0 is the Lie

algebra of G. When G is a torus, the completely integrable actions, i.e.,

those whose nonempty reduced spaces, ^~l{a)/G) are single points,

were classified by Delzant [De]; they all turn out to be Kahler toric

varieties. (Also see [G]). The lowest dimensional Hamiltonian actions

that are not completely integrable are circle actions on 4-manifolds.

M. Audin [Aul, Au2] and K. Ahara and A. Hattori [AH] proved that

for every compact four dimensional Hamiltonian 51-space, the under-

lying manifold and circle action are obtained from a minimal model

by a sequence of equivariant blow ups at fixed points, and they listed

the minimal models. This provides a full list of the four dimensional

manifolds and circle actions that admit symplectic forms and moment

maps. This result is strong and beautiful, but it only answers a small

part of the classification problem. For instance, the above mentioned

authors did not determine which different blowups produce spaces that

are equivariantly diffeomorphic (their list contains repetitions; see our

Examples 3.10, 6.7, and 6.11), nor did they specify which symplectic

forms can be put on these spaces.

The present paper answers these questions. We give a complete

classification of the compact four dimensional Hamiltonian S^-spaces.

The classification consists of a uniqueness part (sections 2-4) and an

existence part (sections 5-7). The uniqueness part tells us how to

determine whether two spaces are isomorphic. The existence part lists

all the possible spaces.

Received by the editor October 30, 1995, and in revised form April 6, 1998.

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