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Journal of Topology 2009 2(3):589-623; doi:10.1112/jtopol/jtp023
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© 2009 London Mathematical Society

Some 6-dimensional Hamiltonian S1-manifolds

Dusa McDuff

Department of Mathematics, Barnard College, Columbia University, New York, 10027-6598 USA dmcduff@barnard.edu


  Abstract

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 Formula by using a new way to desingularize orbifold blow-ups Z of the weighted projective space Formula . 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 S1-manifolds. We also show that (modulo scaling) 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.

Received September 20, 2008. Revised June 26, 2009.

2000 Mathematics Subject Classification 53D05, 53D20, 57S05, 14J30

This work was partially supported by National Science Foundation grant DMS 0604769.


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