# Geometria em Lisboa Seminar

### 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 $\mathcal{S}$.

In this talk we will discuss 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 $\mathbb{C}P^n$.

Current organizer: Rosa Sena Dias