[Paper Review] Floer cohomology of torus fibers and real lagrangians in Fano toric manifolds

Garrett Alston, Lino Amorim|arXiv (Cornell University)|Mar 18, 2010

Geometric and Algebraic Topology15 references11 citations

TL;DR

This paper establishes a combinatorial description of Floer cohomology between torus fibers and the real Lagrangian in Fano toric manifolds using Z2 coefficients and a twisted Floer complex with a locally constant sheaf over F2. It proves that the Floer cohomology HF(R, (Lc, Lρ)) is non-zero if and only if the potential function ∇Wc(ρ) = 0, linking the non-vanishing of HF(R, (Lc, Lρ)) to the critical point condition of the toric potential function, and applies this to prove non-displaceability and lower bounds on intersection points under Hamiltonian isotopy.

ABSTRACT

In this article, we consider the Floer cohomology (with $\Z_2$ coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. We show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. We use this result to develop some applications to non-displaceability and the minimum number of intersection points under Hamiltonian isotopy.

Motivation & Objective

• Understand the conditions under which Floer cohomology between a torus fiber and the real Lagrangian in a Fano toric manifold is well-defined.
• Develop a purely combinatorial description of the Floer complex using the moment polytope and a locally constant sheaf with values in F2.
• Establish a precise criterion—via the potential function Wc—for the non-vanishing of HF(R, (Lc, Lρ)) in terms of critical points of Wc.
• Use the combinatorial description to derive lower bounds on the number of intersection points between the real Lagrangian and Hamiltonian isotopes of torus fibers.
• Prove non-displaceability results for the real Lagrangian and torus fibers using auxiliary constructions in product manifolds and algebraic closure of F2.

Proposed method

• Use the moment polytope of the Fano toric manifold to define a combinatorial model for the Floer complex.
• Twist the standard Floer cochain complex by a locally constant sheaf Lρ with coefficients in F2 to model the Lagrangian pair (R, Lc).
• Count Maslov index 1 holomorphic strips connecting intersection points of R and Lc to define the differential.
• Apply the general Floer theory framework from [9] to define the cohomology, then prove it agrees with the combinatorial count.
• Use the algebraic closure of F2 to ensure existence of solutions to Wc(ρ) = 0 and ∇Wc(ρ) = 0, enabling non-displaceability results.
• Construct a product manifold (XP × XP) to define HF(R × R, (Lc × Lc, Lρ ⊕ Lρ)) when HF(R, (Lc, Lρ)) is not defined, to recover non-displaceability.

Experimental results

Research questions

• RQ1 Under what conditions is the Floer cohomology HF(R, (Lc, Lρ)) well-defined for a torus fiber Lc and the real Lagrangian R in a Fano toric manifold?
• RQ2 How can the Floer complex between R and Lc be described in purely combinatorial terms using the moment polytope?
• RQ3 What is the precise relationship between the non-vanishing of HF(R, (Lc, Lρ)) and the critical points of the potential function Wc?
• RQ4 How does the combinatorial description lead to lower bounds on the number of intersection points between R and Hamiltonian isotopes of Lc?
• RQ5 Can non-displaceability of R and Lc be established even when HF(R, (Lc, Lρ)) is not defined, using auxiliary constructions?

Key findings

• The Floer cohomology HF(R, (Lc, Lρ)) is well-defined if and only if the potential function satisfies Wc(ρ) = 0.
• HF(R, (Lc, Lρ)) is non-zero if and only if ∇Wc(ρ) = 0, establishing a direct link between the cohomology and critical points of Wc.
• For any Hamiltonian diffeomorphism φ such that φ(RP^k) and T^k intersect transversely, the number of intersection points satisfies ♯(φ(RP^k) ∩ T^k) ≥ 2⌊k/2⌋.
• When ∇Wc(ρ) = 0 but Wc(ρ) ≠ 0, HF(R, (Lc, Lρ)) is not defined, yet R and Lc remain non-displaceable.
• In the monotone case, there exists a unique monotone torus fiber L0 such that ∇W0(ρ) = 0 for some Lρ, implying R and L0 are non-displaceable: φ(R) ∩ L0 ≠ ∅ for all Hamiltonian φ.
• Using the product manifold construction, non-displaceability is recovered via HF(R × R, (Lc × Lc, Lρ ⊕ Lρ)) when the original cohomology is not defined, with the rank of this cohomology providing a lower bound on intersection points.