Let $(x,y)$ be the coordinate of $\mathbb{C}^2 \subset \mathbb{P}^1\times \mathbb{P}^1$. Is the $S^1$ action on $\mathbb{P}^1\times \mathbb{P}^1$ given by $$ t\cdot(x,y)=(tx,t^{-1}y) $$ Hamiltonian? Likewise let $(x,y)$ be the coordinate of $\mathbb{C}^2 \subset \mathbb{P}^2$. Is the same action Hamiltonian on $\mathbb{P}^2$?
Here we consider toric symplectic structures and naturally extend the above $T^1$-action on the compactification.