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

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.