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

## TL;DR

This paper investigates Hamiltonian circle actions on symplectic manifolds with isolated fixed points, introducing an equivariant pseudo-index concept, providing bounds, and characterizing the case of maximal pseudo-index as homotopy equivalent to complex projective space.

## Contribution

It defines the equivariant pseudo-index for toric 1-skeletons and establishes upper bounds, characterizing manifolds with maximal pseudo-index.

## Key findings

- Pseudo-index is at most n+1 for certain symplectic manifolds.
- Maximal pseudo-index implies the manifold is homotopically equivalent to P^n.
- Provides bounds for equivariant pseudo-index in Hamiltonian S^1-spaces.

## 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 \(\C P^n\).