Quantum cohomology and Floer invariants of semiprojective toric manifolds

Abstract

We use Floer theory to describe invariants of symplectic $\mathbb{C}^*$-manifolds admitting several commuting $\mathbb{C}^*$-actions. The $\mathbb{C}^*$-actions induce filtrations by ideals on quantum cohomology, as well as filtrations on Hamiltonian Floer cohomologies, and we prove relationships between these filtrations. We also carry this out in the equivariant setting, in particular $\mathbb{C}^*$-actions then give rise to Hilbert-Poincar\'{e} polynomials on ordinary cohomology that depend on Floer theory. For semiprojective toric manifolds, we obtain an explicit presentation for quantum and symplectic cohomology in the Fano and CY setting, both in the equivariant and non-equivariant setting.