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

Abstract: 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 (\C P^n).