In sections 2-4 we associate a labeled graph to every compact four
dimensional Hamiltonian S'1-space, and we show that two such spaces
are isomorphic if and only if they have the same graph. One bi-product
is that a Hamiltonian S^-space is determined up to equivariant sym-
plectomorphism by its underlying manifold, the
action, and the
cohomology class of the symplectic form; see Proposition 4.1.
In section 5 we show that if the fixed points are isolated then the
circle action extends to an action of a two dimensional torus to yield a
toric variety. This leads to a classification of the compact four dimen-
whose fixed points are isolated.
In section 6 we prove that a compact four dimensional Hamiltonian
S1 space can be obtained from a minimal model by a sequence of equi-
variant symplectic blow-ups. This can also be deduced from the similar
result of Audin, Ahara, and Hattori for the underlying S^-manifold, but
our proof is simpler.
Unlike general symplectic blow-ups, the blow-up of a four dimen-
sional Hamiltonian Sfl-space is unambiguous; the resulting Hamiltonian
S^-space is determined up to isomorphism by the fixed point at which
we blow up and the amount by which we blow up. See Proposition 6.1.
To complete the classification it remains to determine by which
amounts it is possible to blow up a Hamiltonian S^-space. Equiva-
lent^, we need to determine which invariant symplectic forms can be
put on our manifolds. This we do in section 7. By the results of sec-
tion 4, it is enough to specify the cohomology classes of the invariant
symplectic forms. Now, it is easy to state a necessary condition for a
cohomology class to represent an invariant symplectic form, and using
Nakai's criterion we show that this condition is also sufficient. More-
over, a cohomology class that satisfies this condition is represented
by a compatible Kdhler form. Hence every compact four dimensional
Hamiltonian Sfl-space is Kahler!
These phenomena do not occur in higher dimensions. S. Tolman
has constructed a compact symplectic 6-manifold with a Hamiltonian
action of a 2-dimensional torus, with isolated fixed points, that does
not admit a compatible Kahler structure. In particular, in dimension
greater than four, having isolated fixed points does not imply that the
space is toric. See [T]; also see [W].
Unfortunately, the methods in this paper do not tell whether two
given Hamiltonian S^-spaces are non-equivariantly symplectomorphic.
In particular, given a symplectic 4-manifold, we do not give the list of
all Hamiltonian S^-actions on it.