Global Hamiltonian Stability of ${\mathrm U}(1)^n$-orbits in $\mathbb P^n(\mathbb C)$

We prove that any ${\mathrm U}(1)^n$-orbit in $\mathbb P^n$ is global Hamiltonian stable. The idea of the proof is the following: (1) we extend one ${\mathrm U}(1)^n$-orbit to the moment torus “fibration” $\{T_t\}_{t\in\Delta_n}$ and consider its Hamiltonian deformation $\{\phi(T_t)\}_{t\in\Delta_n}$ where $\phi$ is a Hamiltonian diffeomorphism of $\mathbb P^n$ and then: (2) we compare each ${\mathrm U}(1)^n$-orbit and its Hamiltonian deformation by looking at the large $k$ asymptotic behavior of the sequence of projective embeddings defined, for each $k$, by the basis of $H^0(\mathbb P^n,\mathcal O(k))$ obtained by the Borthwick–Paul–Uribe semi-clasasical approximation of the $\mathcal O(k)$ Bohr–Sommerfeld tori of the Lagrangian torus fibrations $\{T_t\}_{t\in\Delta_n}$ and its Hamiltoniasn deformation $\{\phi(T_t)\}_{t\in\Delta_n}$.