Some 6-dimensional Hamiltonian S¹ manifolds

In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \CP^2 by using a new way to desingularize orbifold blow ups Z of the weighted projective space \CP^2_{1,m,n}. We now use a related method to construct symplectomorphisms of these spaces Z. This allows us to construct some well known Fano 3-folds (including the Mukai--Umemura 3-fold) in purely symplectic terms, using a classification by Tolman of a particular class of Hamiltonian S^1-manifolds. We also show that these manifolds are uniquely determined by their fixed point data up to equivariant symplectomorphism. As part of this argument we show that the symplectomorphism group of a certain weighted blow up of a weighted projective plane is connected.