[Paper Review] Floer theory and flips

François Charest, Chris Woodward|arXiv (Cornell University)|Aug 7, 2015

Geometric and Algebraic Topology110 references18 citations

TL;DR

This paper establishes that blow-ups and reverse flips in the minimal model program (MMP) for rational symplectic manifolds generate Floer-non-trivial Lagrangian tori. By deforming symplectic structures along the anticanonical class and analyzing Floer cohomology via perturbed pseudoholomorphic curves, the authors show that such surgeries produce Lagrangians with non-vanishing Floer cohomology, providing a symplectic analog to the Bondal-Orlov and Kawamata theorems on derived categories.

ABSTRACT

We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. As applications, we demonstrate the existence of Hamiltonian non-displaceable Lagrangian tori in, for example, small symplectic blow-ups of compact symplectic manifolds and moduli spaces of polygons. These results are part of a conjectural description of generators for the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program.

Motivation & Objective

• To construct Floer-non-trivial Lagrangian tori in rational symplectic manifolds via symplectic surgeries from the minimal model program (MMP).
• To extend the analogy between the decomposition of the bounded derived category of coherent sheaves under MMP transitions and a conjectural decomposition of the Fukaya category.
• To demonstrate that Floer cohomology remains non-trivial under symplectic blow-ups and reverse flips, using perturbed pseudoholomorphic curves and homotopy-theoretic techniques.
• To provide a systematic method for identifying generators of the Fukaya category via mmp runnings in symplectic topology.

Proposed method

• Utilizes a symplectic analog of the minimal model program, where symplectic blow-ups and flips are defined with varying symplectic forms along the anticanonical direction.
• Applies the theory of $A_∞$-algebras and treed pseudoholomorphic disks to compute Floer cohomology and ensure transversality and compactness.
• Employs Hamiltonian perturbations and clean intersections to define and stabilize Floer-theoretic invariants under deformation.
• Uses quilted pseudoholomorphic disks and multiplihedra to construct $A_∞$-morphisms and homotopies between Fukaya algebras.
• Introduces the concept of 'broken' symplectic manifolds and perturbation data to model degenerations and transitions in the mmp.
• Applies stabilization techniques and infinite-length limits to analyze the behavior of Floer cohomology across mmp transitions.

Experimental results

Research questions

• RQ1 Can symplectic blow-ups and flips produce Lagrangian tori with non-vanishing Floer cohomology in rational symplectic manifolds?
• RQ2 How does the Fukaya category of a compact symplectic manifold decompose under a running of the minimal model program, analogous to the derived category decomposition in algebraic geometry?
• RQ3 What is the role of the anticanonical direction in deforming symplectic structures to generate non-trivial Floer cohomology?
• RQ4 How do Floer cohomology invariants behave under symplectic surgeries such as flips and blow-ups, and what structures govern their invariance?
• RQ5 Can the location and non-triviality of Floer-non-trivial tori be predicted via moment map data or labellings in toric or polygon spaces?

Key findings

• Blow-ups or reverse flips of rational symplectic manifolds with point centers produce Lagrangian tori with non-vanishing Floer cohomology.
• For toric manifolds, the moment map fibers over singular points in the mmp running become Floer-non-trivial for small time offsets before the critical time.
• In polygon spaces, regular labellings of triangulated polygons yield Lagrangian tori with non-trivial Floer cohomology under mmp transitions.
• For moduli spaces of flat $SU(2)$-bundles on punctured spheres, the Goldman Lagrangian over a regular labelling has non-trivial Floer cohomology when the looseness parameter is sufficiently small.
• The Floer cohomology remains non-trivial for finite times before the next mmp transition, as shown by Palmer-Woodward's results.
• Standard Lagrangian tori in Darboux charts have trivial Floer cohomology despite non-empty Maurer-Cartan space, highlighting the non-triviality of the constructed examples.