Hamiltonian S^1-spaces with large equivariant pseudo-index

Let $(M,\omega )$ be a compact symplectic manifold of dimension $2n$ endowed with a Hamiltonian circle action with only isolated fixed points. Whenever $M$ admits a toric $1$-skeleton $\mathcal{S}$, which is a special collection of embedded $2$-spheres in $M$, we define the notion of equivariant pseudo-index of $\mathcal{S}$: this is the minimum of the evaluation of the first Chern class $c_1$ on the spheres of $\mathcal{S}$. This can be seen as the analog in this category of the notion of pseudo-index for complex Fano varieties. In this paper we provide upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of $M$ are unimodal, we prove that it is at most $n+1$ . Moreover, when it is exactly $n+1$, $M$ must be homotopically equivalent to $\text{C}{P}^n$.