If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic? Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?
 A: I think not, my argument is slightly messy but it works I guess.
Take the Hirzebruch surface $S=\mathbb{F}_2$ with its standard toric action, it is diffeomorphic to $S'=\mathbb{CP}^1 \times \mathbb{CP}^1$, but I claim they are not equivariantly diffeomorphic (with respect to the standard toric action on $\mathbb{CP}^1 \times \mathbb{CP}^1$).
Suppose that they are, so $F: S \rightarrow S'$ is an equivariant diffeomorphism.
Then, pick a subtorus $S^1 \subset T^2 $ such that the corresponding Hamiltonian $S^1$-action on $S$ has an isotropy sphere $C'$ with $C' \cdot C' = 2$ (it will be one of the sections of the $\mathbb{CP}^1$-bundle). Then, there will be a corresponding subtorus action on $S'$, also having an isotropy curve $C$ with $C \cdot C = 2$. But the isotropy submanifolds for subtorus actions on toric surfaces are always the boundary divisors so $C \cdot C = 0$, a contradiction (because all of the four boundary divisors of S' are factors having self-intersection $0$).
For reference, an isotropy submanifold is a connected component of the set $\{p \in M : z. p = p \;\; \forall  z \in \mathbb{Z}_{k} \subset S^1 \}$ for some $k>1$. it is a purely topological notion (not depending on the almost complex structure, symplectic form, etc.) hence pulls back via an equivariant diffeomorphism.
