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Hamiltonian circle actions with fixed point set almost minimal

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Abstract

Motivated by recent works on Hamiltonian circle actions satisfying certain minimal conditions, in this paper, we consider Hamiltonian circle actions satisfying an almost minimal condition. More precisely, we consider a compact symplectic manifold \((M, \omega )\) admitting a Hamiltonian circle action with fixed point set consisting of two connected components X and Y satisfying \(\dim (X)+\dim (Y)=\dim (M)\). Under certain cohomology conditions, we determine the circle action, the integral cohomology rings of M, X and Y, and the total Chern classes of M, X, Y, and of the normal bundles of X and Y. The results show that these data are unique—they are exactly the same as those in the standard example \(\widetilde{G}_2({\mathbb {R}}^{2n+2})\), the Grassmannian of oriented 2-planes in \({\mathbb {R}}^{2n+2}\), which is of dimension 4n with (any) \(n\in {\mathbb {N}}\), equipped with a standard circle action. Moreover, if M is Kähler and the action is holomorphic, we can use a few different criteria to claim that M is \(S^1\)-equivariantly biholomorphic and \(S^1\)-equivariantly symplectomorphic to \(\widetilde{G}_2({\mathbb {R}}^{2n+2})\).

Keywords

Symplectic manifold Hamiltonian circle action Equivariant cohomology Characteristic classes Kähler manifold Symplectomorphism Biholomorphism 

Mathematics Subject Classification

53D05 53D20 55N25 57R20 32H02 

Notes

Acknowledgements

The author would like to thank the referee for some comments which help to improve the exposition. This work is supported by the NSFC grant K110712116.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of mathematical SciencesSoochow UniversitySuzhouChina

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