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Log Floer cohomology for oriented log symplectic surfaces

Charlotte Kirchhoff-Lukat

TL;DR

This work defines log Floer cohomology for orientable log symplectic surfaces, extending Lagrangian intersection Floer theory to Poisson structures with a degeneracy locus Z. It constructs the log Floer complex using admissible Hamiltonians to handle intersections with Z, introduces crossing lunes for multi-component cases, and proves the differential squares to zero and invariance under Z-fixed isotopies. For a single Lagrangian, the theory recovers log de Rham cohomology relative to α∩Z, while for pairs of Lagrangians it yields a combinatorial, invariant count of intersections that accounts for the degeneracy locus. The paper also discusses a Novikov-field refinement and derives the log-Floer equation as a natural pseudoholomorphic formulation, setting the stage for higher A∞-structures and the Fukaya category in the accompanying KL23 article. This work thus provides a rigorous foundation for Floer-type invariants on log Poisson/symplectic surfaces, with potential implications for generalized complex geometry and mirror symmetry in non-symplectic settings.

Abstract

This article provides the first extension of Lagrangian Intersection Floer cohomology to Poisson structures which are almost everywhere symplectic, but degenerate on a lowerdimensional submanifold. The main result of the article is the definition of Lagrangian intersection Floer cohomology, referred to as log Floer cohomology, for orientable surfaces equipped with log symplectic structures. We show that this cohomology is invariant under suitable isotopies and that it is isomorphic to the log de Rham cohomology when computed for a single Lagrangian.

Log Floer cohomology for oriented log symplectic surfaces

TL;DR

This work defines log Floer cohomology for orientable log symplectic surfaces, extending Lagrangian intersection Floer theory to Poisson structures with a degeneracy locus Z. It constructs the log Floer complex using admissible Hamiltonians to handle intersections with Z, introduces crossing lunes for multi-component cases, and proves the differential squares to zero and invariance under Z-fixed isotopies. For a single Lagrangian, the theory recovers log de Rham cohomology relative to α∩Z, while for pairs of Lagrangians it yields a combinatorial, invariant count of intersections that accounts for the degeneracy locus. The paper also discusses a Novikov-field refinement and derives the log-Floer equation as a natural pseudoholomorphic formulation, setting the stage for higher A∞-structures and the Fukaya category in the accompanying KL23 article. This work thus provides a rigorous foundation for Floer-type invariants on log Poisson/symplectic surfaces, with potential implications for generalized complex geometry and mirror symmetry in non-symplectic settings.

Abstract

This article provides the first extension of Lagrangian Intersection Floer cohomology to Poisson structures which are almost everywhere symplectic, but degenerate on a lowerdimensional submanifold. The main result of the article is the definition of Lagrangian intersection Floer cohomology, referred to as log Floer cohomology, for orientable surfaces equipped with log symplectic structures. We show that this cohomology is invariant under suitable isotopies and that it is isomorphic to the log de Rham cohomology when computed for a single Lagrangian.
Paper Structure (14 sections, 12 theorems, 52 equations, 18 figures)

This paper contains 14 sections, 12 theorems, 52 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Orientable log symplectic surfaces
  4. Classification
  5. Anatomy
  6. Compact Lagrangians and Hamiltonian isotopies
  7. Log Floer cohomology for log symplectic surfaces
  8. Surface with boundary and a single symplectic component
  9. Closed surface with multiple symplectic components
  10. What can we say about the log Floer cohomology of different Lagrangians in this setting?
  11. A brief aside: Working over the Novikov field
  12. The Floer equation for log symplectic Floer theory and (pseudo-)holomorphic lunes
  13. Outlook
  14. Cases in the proof of Theorem \ref{['prop:sym']}

Key Result

Theorem 1

(Theorem thm:full) For $\alpha,\beta\subset (M,\omega,Z)$ closed compact Lagrangians which intersect the degeneracy locus $Z$ transversely and are not null-homotopic under path homotopy leaving $Z$ invariant, the pair $(CF(\alpha,\phi_H(\beta),\partial))$ with $H$ an admissible Lagrangian and $\part

Figures (18)

  • Figure 1: Preferred local coordinates around $Z_i$
  • Figure 2: With a chosen global orientation of $S$, the sign of the function $h$ determines the orientation of $all$ boundary circles. Here we illustrate the two possible configurations.
  • Figure 3: Illustration of the path between two boundary circles $Z_0,Z_1$ of $S$ together with chosen coordinate neighbourhoods and a graph of the function $h$ near the endpoints of $\gamma$. The dotted section of the graph indicates that the precise behaviour of $h$ is unknown in this section; we only know that $h>0$.
  • Figure 4: Examples of possible zero loci of log symplectic structures on the orientable surface of genus two, together with the corresponding bipartite graphs. Here, $g$ denotes the genus of the graph. For each example, one of the possible vertex 2-colourings is shown, corresponding to one of the possible relative orientation of the log symplectic structure and global orientation.
  • Figure 5: Degrees of intersection points of Lagrangians when $h>0$.
  • ...and 13 more figures

Theorems & Definitions (47)

  • Theorem
  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 37 more