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Quantum cohomology and Floer invariants of semiprojective toric manifolds

Alexander F. Ritter, Filip Živanović

TL;DR

The paper extends Floer-theoretic constructions to non-compact symplectic manifolds with commuting $\mathbb{C}^*$-actions, introducing filtrations on $QH^*(Y)$ and $SH^*(Y,\varphi)$ that behave well under multiple actions and their localisations. It develops an equivariant projective-morphism framework, including the $\Upsilon$ map and rotation classes, to compare filtrations across actions and define a universal core for contracting torus actions. Specialising to semiprojective toric manifolds, the authors derive explicit presentations of quantum and symplectic cohomologies in the Fano and CY settings, identifying $SH^*(Y,\varphi)$ with the Jacobian ring $\mathrm{Jac}(W)$ after localisation at rotation classes. A detailed toric example shows how non-convex (non-convex-at-infinity) cases can exhibit vanishing symplectic cohomology after localisation, while retaining a robust quantum-structure. Overall, the work broadens the scope of Seidel-type and localisation phenomena to a wide class of non-compact toric spaces, with concrete algebraic presentations and computable filtrations that illuminate the interplay between toric geometry, Floer theory, and quantum cohomology.

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-Poincaré 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.

Quantum cohomology and Floer invariants of semiprojective toric manifolds

TL;DR

The paper extends Floer-theoretic constructions to non-compact symplectic manifolds with commuting -actions, introducing filtrations on and that behave well under multiple actions and their localisations. It develops an equivariant projective-morphism framework, including the map and rotation classes, to compare filtrations across actions and define a universal core for contracting torus actions. Specialising to semiprojective toric manifolds, the authors derive explicit presentations of quantum and symplectic cohomologies in the Fano and CY settings, identifying with the Jacobian ring after localisation at rotation classes. A detailed toric example shows how non-convex (non-convex-at-infinity) cases can exhibit vanishing symplectic cohomology after localisation, while retaining a robust quantum-structure. Overall, the work broadens the scope of Seidel-type and localisation phenomena to a wide class of non-compact toric spaces, with concrete algebraic presentations and computable filtrations that illuminate the interplay between toric geometry, Floer theory, and quantum cohomology.

Abstract

We use Floer theory to describe invariants of symplectic -manifolds admitting several commuting -actions. The -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 -actions then give rise to Hilbert-Poincaré 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.
Paper Structure (36 sections, 54 theorems, 189 equations)

This paper contains 36 sections, 54 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. A brief overview of the structural properties of invariants for symplectic $\mathbb C^*$-manifolds
  4. Examples of algebraic torus actions on symplectic $\mathbb C^*$-manifolds
  5. Equivariant projective morphisms
  6. Semiprojective toric manifolds
  7. Commuting, complete, and contracting $\mathbb C^*$-actions on affine varieties
  8. Complete and contracting $\mathbb C^*$-actions on affine varieties
  9. Commuting $\mathbb C^*$-actions
  10. Torus actions
  11. Further technical properties of $N_0(X)$ and $N_+(X)$
  12. Commuting $\mathbb C^*$-actions on symplectic $\mathbb C^*$-manifolds
  13. Symplectic $\mathbb C^*$-actions and the Core
  14. The semigroup of contracting actions $N_+(Y)$ of an algebraic torus action
  15. Equivariant projective morphisms
  16. ...and 21 more sections

Key Result

Theorem 1.2

The homomorphism $c^*:QH^*(Y)\to SH^*(Y,\varphi)$ is surjective. It equals localisation at $Q_{\varphi}\in QH^{2\mu}(Y),$ where $\mu$ is the Maslov index of the $S^1$-action $\varphi$. where $E_0(Q_{\varphi})=\ker c^* \subset QH^*(Y)$ is the generalised $0$-eigenspace of quantum product by $Q_{\varphi}$. For $N^+\in \mathbb R$ just above $N\in \mathbb N$, the continuation map $c^*_{N^+}$ (whose d

Theorems & Definitions (136)

  • Remark 1.1
  • Theorem 1.2: RZ1
  • Example 1.3: R16
  • Theorem 1.4: RZ1RZ2
  • Definition 1.5
  • Lemma 1.6: RZ1RZ3
  • Theorem 1.7: Localisation and Injectivity Theorems, RZ3
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 126 more