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arXiv · 1311.2327

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

Garrett Alston
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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).

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Garrett Alston. 2013-11-11. Floer cohomology of immersed Lagrangian spheres in smoothings of $A_N$ surfaces. https://arxiv.org/abs/1311.2327

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