On the dynamics characterization of complex projective spaces

TL;DR

This paper proves that certain symplectic manifolds with specific topological and dynamical properties are necessarily monotone and share quantum homology with complex projective space, with a special case identifying them as .

Contribution

It establishes a characterization of symplectic manifolds with Hamiltonian toric pseudo-rotations, linking their topology and quantum homology to complex projective spaces.

Findings

Manifolds are necessarily monotone under given conditions.

Quantum homology is isomorphic to that of .

When n=2, the manifold is symplectomorphic to .

Abstract

We show that a closed weakly-monotone symplectic manifold of dimension $2n$ which has minimal Chern number greater than or equal to $n+1$ and admits a Hamiltonian toric pseudo-rotation is necessarily monotone and its quantum homology is isomorphic to that of the complex projective space. As a consequence when $n=2$, the manifold is symplectomorphic to $\mathbb{C}P^2$.