Hamiltonian circle actions with fixed point set almost minimal

Motivated by recent works on Hamiltonian circle actions satisfying certain minimal conditions, in this paper, we consider Hamiltonian circle actions satisfying an almost minimal condition. More precisely, we consider a compact symplectic manifold $(M, \omega)$ admitting a Hamiltonian circle action with fixed point set consisting of two connected components $X$ and $Y$ satisfying $\dim(X)+\dim(Y)=\dim(M)$. Under certain cohomology conditions, we determine the circle action, the integral cohomology rings of $M$, $X$ and $Y$, and the total Chern classes of $M$, $X$, $Y$, and of the normal bundles of $X$ and $Y$. The results show that these data are unique --- they are exactly the same as those in the standard example $\Gt_2(\R^{2n+2})$, the Grassmannian of oriented $2$-planes in $\R^{2n+2}$, which is of dimension $4n$ with (any) $n\in\N$, equipped with a standard circle action. Moreover, if $M$ is K\"ahler and the action is holomorphic, we can use a few different criteria to claim that $M$ is $S^1$-equivariantly biholomorphic and $S^1$-equivariantly symplectomorphic to $\Gt_2(\R^{2n+2})$.