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Quasi-morphisms on contactomorphism groups and Grassmannians of 2-planes

  • Frol ZapolskyEmail author
Original Paper
  • 9 Downloads

Abstract

We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the contactomorphism group of its total space carries a nonzero homogeneous quasi-morphism. The construction uses Givental’s nonlinear Maslov index and a reduction theorem for quasi-morphisms on contactomorphism groups previously established together with M. S. Borman. We explore applications to metrics on this group and to symplectic and contact rigidity. In particular we obtain a new proof that the quaternionic projective space \({\mathbb {H}}P^{n-1}\), naturally embedded in the Grassmannian \({{\,\mathrm{G}\,}}_2({\mathbb {C}}^{2n})\) as a Lagrangian, cannot be displaced from the real part \({{\,\mathrm{G}\,}}_2({\mathbb {R}}^{2n})\) by a Hamiltonian isotopy.

Keywords

Quasi-morphism Symplectic geometry Contact geometry Contactomorhism group Grassmannian Contact rigidity Symplectic rigidity 

Mathematics Subject Classification (2000)

53D20 20F38 

Notes

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Natural SciencesUniversity of HaifaHaifaIsrael

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