[Paper Review] Remarks on Lagrangian intersections in toric manifolds

Miguel Abreu, Leonardo Macarini|arXiv (Cornell University)|May 3, 2011

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper establishes non-displaceability of Lagrangian torus fibers and their intersections with the real part in toric symplectic manifolds using symplectic reduction and cartesian product constructions. By leveraging rigidity results from weighted projective spaces, the authors extend these to all monotone toric symplectic manifolds and construct examples with continuous families of non-displaceable fibers, including non-Fano and non-monotone cases.

ABSTRACT

We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can use simple cartesian product and symplectic reduction considerations to go from basic examples to much more sophisticated ones. We show in particular how rigidity results for the above Lagrangian intersection problems in weighted projective spaces can be combined with these considerations to prove analogous results for all monotone toric symplectic manifolds. We also discuss non-monotone and/or non-Fano examples, including some with a continuum of non-displaceable torus orbits.

Motivation & Objective

• To establish rigidity results for Lagrangian intersections involving torus orbits and the real part in toric symplectic manifolds.
• To extend non-displaceability results from basic examples—such as weighted projective spaces—to more general monotone and non-monotone toric symplectic manifolds.
• To construct examples with continuous intervals of non-displaceable torus fibers, including non-Fano and non-monotone cases.
• To demonstrate how symplectic reduction and cartesian products can generate sophisticated examples from simple ones.
• To provide a unified framework for understanding non-displaceability in terms of moment polytope geometry and Hamiltonian actions.

Proposed method

• Use of symplectic reduction to construct new toric manifolds from products of simpler ones, preserving non-displaceability of fibers.
• Application of cartesian product constructions to extend non-displaceability results from lower-dimensional examples to higher-dimensional ones.
• Leveraging known rigidity theorems for the central fiber in weighted projective spaces $\mathbb{C}P(1,1,k)$ to infer non-displaceability in derived manifolds.
• Employing moment map descriptions and action-angle coordinates to analyze the structure of torus orbits and their intersections with the real part.
• Utilizing the condition $\sum \nu_i = 0$ on primitive normals to the polytope facets to deduce non-displaceability of the special torus fiber in monotone cases.
• Constructing explicit symplectic reductions with specified level sets to realize target polytopes and locate non-displaceable fibers.

Experimental results

Research questions

• RQ1 Can non-displaceability of the central torus fiber in weighted projective spaces be extended to all monotone toric symplectic manifolds?
• RQ2 What is the role of symplectic reduction and cartesian products in generating non-displaceable Lagrangian torus fibers in complex toric manifolds?
• RQ3 Do non-Fano or non-monotone toric symplectic 4-manifolds admit continuous families of non-displaceable torus fibers?
• RQ4 Under what conditions on the moment polytope does the special torus fiber or its intersection with the real part remain non-displaceable under Hamiltonian isotopies?
• RQ5 How can the real part of a toric manifold and its intersections with torus orbits be systematically analyzed using geometric and topological constraints?

Key findings

• For monotone toric symplectic manifolds with $\sum \nu_i = 0$, the special torus fiber $T$ satisfies $\psi(T) \cap T \neq \emptyset$ and $\sharp(\psi(T) \pitchfork T) \geq 2^n$ for all $\psi \in \operatorname{Ham}(M,\omega)$.
• The real part $R$ of a monotone toric manifold satisfies $\psi(T) \cap R \neq \emptyset$ for all $\psi \in \operatorname{Ham}(M,\omega)$ under the same condition.
• The central torus fiber in $\mathbb{C}P(1,1,k)$ is non-displaceable, and this result extends via symplectic reduction to all Hirzebruch surfaces $H_k$.
• A non-Fano toric 4-manifold with moment polytope defined by five inequalities admits a continuous interval of non-displaceable torus fibers over $x_1 = \lambda$, $x_2 = 0$ with $1 < \lambda < 2$.
• The non-displaceable fibers in the non-Fano example arise as the image of a symplectic reduction of $\mathbb{C}P(1,1,2) \times \mathbb{C}P^1 \times \mathcal{O}(-1)$ at a specific level set.
• The construction yields a continuum of non-displaceable torus fibers in certain non-monotone toric 4-manifolds, demonstrating that non-displaceability is not exclusive to monotone or Fano cases.