On an exotic Lagrangian torus in $\mathbb{C}P^2$

TL;DR

This paper constructs a special Lagrangian torus in complex projective space, demonstrating its non-displaceability and superheaviness, which answers a longstanding question about the structure of $ ext{CP}^2$.

Contribution

It introduces a new non-displaceable Lagrangian torus in $ ext{CP}^2$, showing that $ ext{RP}^2$ is not a stem, advancing understanding of symplectic topology.

Findings

Identifies a non-displaceable Lagrangian torus in $ ext{CP}^2$

Proves $ ext{RP}^2$ is not a stem in $ ext{CP}^2$

Establishes superheaviness with respect to a symplectic quasi-state

Abstract

We find a non-displaceable Lagrangian torus fiber in a semi-toric system, which is superheavy with respect to certain symplectic quasi-state. In particular, this proves Lagrangian $\RR P^2$ is not a stem in $\CC P^2$, answering a question of Entov and Polterovich.