supported by NSF grant DMS-9404404 during part of the work on this
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 (/*,/) =
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).
In section 2.1 we associate a labeled graph to each compact four
dimensional Hamiltonian
space. In section 2.2 we describe the most
Clearly, every complex projective surface is a symplectic four manifold. How-
ever, non-isomorphic complex projective surfaces could be isomorphic as Hamilton-
see Example 6.7 of the present paper. In the other direction, every
compact four dimensional Hamiltonian 5
space is equivariantly symplectomorphic
to a complex projective surface with a holomorphic circle action [AH, Aul]. More-
over, every compact four dimensional Hamiltonian
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.
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