Using Lefschetz fibrations, we construct nonstandard symplectic structures on cotangent bundles of spheres. These structures are of Liouville type, which means exact and convex at infinity.
We study the symplectic topology of some finite algebraic quotients of the A n Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern's rational blowdown construction. We prove that these affine surfaces have no closed 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 other hand, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori which we study in the A n Milnor fibre. We conclude that these affine surfaces have non-vanishing symplectic cohomology.
We study the symplectic topology of some finite algebraic quotients of the A n Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern's rational blowdown construction. We prove that these affine surfaces have no closed 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 other hand, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori which we study in the A n Milnor fibre. We conclude that these affine surfaces have non-vanishing symplectic cohomology.
IntroductionLet p > q > 0 be relatively prime integers. In [11], Casson and Harer introduced rational homology balls B p,q which are bounded by the lens space L(p 2 , pq − 1). These homology balls were subsequently used in Fintushel-Stern's rational blow-down construction [15] (see also, [32]). In fact, B p,q are naturally equipped with Stein structures since they are affine varieties (cf. [23]) and here we are concerned with the symplectic topology of these Stein surfaces.The key topological fact is that B p,q are p-fold covered (without ramification) by the Milnor fibre of the A p−1 singularity. The latter has a unique Stein structure and its symplectic topology is well-studied (see [24], [28], [41], [36]).Following Seidel [35], we make the following definition: Definition 1.1 A Stein manifold X is said to be empty if its symplectic cohomology vanishes. It is non-empty otherwise.We recommend [38] for an excellent survey of symplectic cohomology. Non-empty Stein manifolds are often detected by the following important theorem of Viterbo (here stated in a weak form):
The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, W , of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of W are non-degenerate. In this paper we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.2000 Mathematics Subject Classification. Primary 52B20. Secondary 14N35, 53D45.
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