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arXiv · 1902.02403

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

Frol Zapolsky
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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. Strom 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, naturally embedded in the Grassmannian of 2-planes in a 2n-dimensional complex space as a Lagrangian, cannot be displaced from the real part of the complex Grassmannian by a Hamiltonian isotopy.

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Frol Zapolsky. 2019-02-06. Quasi-morphisms on contactomorphism groups and Grassmannians of 2-planes. https://arxiv.org/abs/1902.02403

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