Hamiltonian S^1-manifolds of dimension 2n with n+2 isolated fixed points

Let $(M, \omega)$ be a compact $2n$-dimensional symplectic manifold equipped with a Hamiltonian $S^1$ action with $n+2$ isolated fixed points. We will see that $n$ must be even. Such an example is $\Gt_2(\R^{n+2})$, the Grassmanian of oriented $2$-planes in $\R^{n+2}$ with $n$ even, equipped with a standard $S^1$ action. We show that if the $S^1$ representations at the fixed points on $M$ are the same as those of the standard $S^1$ action on $\Gt_2(\R^{n+2})$, then the integral cohomology ring and total Chern class of $M$ are the same as those of $\Gt_2(\R^{n+2})$; on the other hand, if $M$ has the same integral cohomology ring as $\Gt_2(\R^{n+2})$, then the $S^1$ representations at the fixed points are the same as those of the standard $S^1$ action on $\Gt_2(\R^{n+2})$. In particular, if $M$ is K\"ahler and the action is holomorphic, then any of the following $3$ conditions implies that $M$ is equivariantly biholomorphic and symplectomorphic to $\Gt_2(\R^{n+2})$: (1) $M$ has the same first Chern class as $\Gt_2(\R^{n+2})$, (2) $M$ has the same integral cohomology ring as $\Gt_2(\R^{n+2})$, and (3) the $S^1$ representations at the fixed points are the same as those of the standard $S^1$ action on $\Gt_2(\R^{n+2})$.