2

YAEL KARSHON

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

S1

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-

sional Hamiltonian

51-spaces

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.