Log Floer cohomology for oriented log symplectic surfaces

TL;DR

This paper extends Lagrangian Floer cohomology to log symplectic surfaces, providing a new invariant for Poisson structures that are symplectic almost everywhere but degenerate on a submanifold.

Contribution

It introduces log Floer cohomology for orientable log symplectic surfaces and proves its invariance and isomorphism to log de Rham cohomology in certain cases.

Findings

Log Floer cohomology is invariant under isotopies.

Log Floer cohomology is isomorphic to log de Rham cohomology for a single Lagrangian.

First extension of Floer cohomology to almost everywhere symplectic Poisson structures.

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.