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Foundations of Computational Mathematics

, Volume 14, Issue 4, pp 791–860 | Cite as

New Tools for Classifying Hamiltonian Circle Actions with Isolated Fixed Points

  • Leonor Godinho
  • Silvia Sabatini
Article

Abstract

For every compact almost complex manifold \((\mathsf {M},\mathsf {J})\) equipped with a \(\mathsf {J}\)-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that \(\mathsf {M}\) is symplectic and the action is Hamiltonian. If the manifold satisfies an extra so-called positivity condition, then this algorithm determines a family of vector spaces that contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever \(\dim (\mathsf {M})\le 6\) and, when \(\dim (\mathsf {M})=8\), whenever the \(S^1\)-action extends to an effective Hamiltonian \(T^2\)-action, or none of the isotropy weights is \(1\). Moreover, there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces. We run the algorithm for \(\dim (\mathsf {M})\le 8\), quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for \(\dim (\mathsf {M})=6\) and, when \(\dim (\mathsf {M})=8\), we prove that the equivariant cohomology ring, Chern classes, and isotropy weights agree with those of \({\mathbb {C}}P^4\) with the standard \(S^1\)-action (thereby proving the symplectic Petrie conjecture in this setting).

Keywords

Circle actions Fixed points Equivariant cohomology 

Mathematics Subject Classification

53D20 19J35 37B05 

Notes

Acknowledgments

We thank Tudor Ratiu for his support, Susan Tolman for introducing us to this problem, Victor Guillemin for useful discussions, and Manuel Racle, José Braga, and Carlos Henriques and the students Filipe Casal, Francisco Pavão Martins, and Diogo Poças for helping us with our first steps in C++ and Mathematica. This work was partially supported by Fundação para a Ciência e Tecnologia (FCT/Portugal)

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Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.Departamento de Matemática, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos-LARSYS, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

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