Quantum cohomology and Floer invariants of semiprojective toric manifolds

TL;DR

This paper explores Floer theory to describe invariants of semiprojective toric manifolds with multiple commuting a7-actions, providing explicit presentations of quantum and symplectic cohomology in various settings.

Contribution

It introduces a novel relationship between filtrations induced by a7-actions on quantum and Floer cohomologies, with explicit results for semiprojective toric manifolds.

Findings

Relationships between filtrations on quantum and Floer cohomologies established

Explicit presentations of quantum and symplectic cohomology for semiprojective toric manifolds

Dependence of Hilbert-Poincare9 polynomials on Floer theory in equivariant setting

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.