2008/08/31 by Dusa McDuff · 1 citation
Mathematics · #Disjoint sets #Embedding #Equivariant map #Fano plane #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Linear subspace #Orbifold #Projective plane #Symplectic geometry #Symplectomorphism #math.DG #math.SG #msc:14J30 #msc:53D05 #msc:53D20 #msc:57S05
paper · pdf · doi:10.1112/jtopol/jtp023
40 pages, 9 figures; v2: proof of Thm 2.16 improved, figure added; to be published in Journal of Topology
openalex publication_date 2009/01/01 · arxiv created 2009/07/29 · arxiv updated 2014/02/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 ℂP2 by using a new way to desingularize orbifold blow-ups Z of the weighted projective space ℂP21,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 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.