4

YAEL KARSHON

supported by NSF grant DMS-9404404 during part of the work on this

paper.

Related works: Most important is the work of Michele Audin

[Aul, Au2] and of K. Ahara and A. Hattori [AH] that was mentioned

earlier. T. Delzant [De] classified the completely integrable Hamilton-

ian torus actions on compact symplectic manifolds. He also showed

that a Hamiltonian circle action whose moment map has exactly two

singular values, one of which is non-degenerate, is isomorphic to a

standard action on CFn. For Hamiltonian circle actions on compact

symplectic four manifolds, already in 1989 E. Lerman wrote explicit

formulas describing the pre-image via the moment map of an interval

of regular values [LI].

In other than the symplectic category, locally smooth circle actions

on four manifolds (no symplectic structure) were classified by Fintushel

[Fin], and holomorphic circle actions on complex projective surfaces

were classified by Orlik and Wagreich [OW], who produced a list of

surfaces and actions identical to Audin's list in [Aul].1

Hamiltonian actions of non-abelian groups are more difficult than

torus actions. Completely integrable actions of non-abelian groups have

been studied in recent years by Delzant, Guillemin, Knop, Sjamaar, de

Souza, Woodward, and possibly others.

A technical remark: We use the following conventions. The

circle group consists of the complex numbers of norm 1. Its Lie algebra

is identified with M such that the exponential map is t i— elt and its

kernel is I = 27rZ. The dual of the Lie algebra is t* = M, and the weight

lattice is I* = Z, so that (/*,/) =

2TTZ.

Lebesgue measure on t* is the

standard measure on R so that the volume of t*/P is 1. The symplectic

form on C is rdr A d6 (in polar coordinates), and the moment map for

the standard circle action is $ = r 2 /2. A disc around the origin of area

27rA gets mapped to an interval of length A (area= 7rr2, A = r 2 /2).

2. GRAPHS

In section 2.1 we associate a labeled graph to each compact four

dimensional Hamiltonian

S1

space. In section 2.2 we describe the most

1

Clearly, every complex projective surface is a symplectic four manifold. How-

ever, non-isomorphic complex projective surfaces could be isomorphic as Hamilton-

ian

51-spaces;

see Example 6.7 of the present paper. In the other direction, every

compact four dimensional Hamiltonian 5

1

space is equivariantly symplectomorphic

to a complex projective surface with a holomorphic circle action [AH, Aul]. More-

over, every compact four dimensional Hamiltonian

S1

space is Kahler; see section

7 of the present paper. Of course, if the two-form is not integral, the space is not

symplectomorphic to a complex projective surface.