Floer cohomology of immersed Lagrangian spheres in smoothings of $A_N$ surfaces

TL;DR

This paper computes the self-Floer cohomology of certain immersed Lagrangian spheres in smoothings of A_N surfaces, revealing cases where the Floer differential vanishes, using explicit holomorphic strip calculations.

Contribution

It provides explicit calculations of Floer cohomology for immersed Lagrangian spheres in A_N surface smoothings, utilizing Lefschetz fibrations for holomorphic strip enumeration.

Findings

Most immersed spheres have zero Floer differential with Z/2 coefficients.

Explicit Floer cohomology calculations are achieved using Lefschetz fibrations.

The work advances understanding of Floer theory in singularity smoothing contexts.

Abstract

We calculate the self-Floer cohomology with Z/2 coefficients of some immersed Lagrangian spheres in the affine symplectic submanifolds of C^3 that are smoothings of A_N surfaces. The immersed spheres are exact and graded. Moreover, they satisfy a positivity assumption that allows us to calculate the Floer cohomology as follows: Given auxiliary data a Morse function on S^2 and a time-dependent almost complex structure, the Floer cochain complex is the Morse complex plus two generators for each self-intersection point of the Lagrangian sphere. The Floer differential is defined by counting combinations of Morse flow lines and holomorphic strips. Using a Lefschetz fibration allows us to explicitly calculate all holomorphic strips and describe the Floer differential. For most of the immersed spheres the Floer differential is zero (with Z/2-coefficients).