Mathematics – Symplectic Geometry
Scientific paper
2012-02-25
Mathematics
Symplectic Geometry
27 pages, 4 figures. A Legendrian surgery diagram is provided in the new Section 3.2
Scientific paper
We compute the Floer cohomology of monotone tori in the Stein surfaces obtained by a linear plumbing of cotangent bundles of spheres, also known as the Milnor fibre associated with the complex surface singularity of type A_n. We next study some finite quotients of the A_n Milnor fibre which coincide with the Stein surfaces that appear in Fintushel and Stern's rational blowdown construction. We show that these Stein surfaces have no exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the A_n Milnor fibre coming from homological mirror symmetry. On the contrary, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori that we studied in the A_n Milnor fibre. We conclude that these Stein surfaces have non-vanishing symplectic cohomology.
Lekili Yanki
Maydanskiy Maksim
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