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
= 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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