From symplectic cohomology to Lagrangian enumerative geometry

We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of LG potentials. We discover a relation between higher disk potentials and symplectic cohomology rings of anticanonical divisor complements (themselves related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains.Comment: 47 pages, 13 figures; v2: reference fixes, minor correction

Floer theory for negative line bundles via Gromov-Witten invariants

Let M be the total space of a negative line bundle over a closed symplectic manifold. We prove that the quotient of quantum cohomology by the kernel of a power of quantum cup product by the first Chern class of the line bundle is isomorphic to symplectic cohomology. We also prove this for negative vector bundles and the top Chern class. We explicitly calculate the symplectic and quantum cohomologies of O(-n) over P^m. For n=1, M is the blow-up of C^{m+1} at the origin and symplectic cohomology has rank m. The symplectic cohomology vanishes if and only if the first Chern class of the line bundle is nilpotent in quantum cohomology. We prove a Kodaira vanishing theorem and a Serre vanishing theorem for symplectic cohomology. In general, we construct a representation of \pi_1(Ham(X,\omega)) on the symplectic cohomology of symplectic manifolds X conical at infinity.Comment: 53 pages; version 3: improved discussion of maximum principle for negative vector bundles. The final version is published in Advances in Mathematic

Higher algebraic structures in Hamiltonian Floer theory I

This is the first of two papers devoted to showing how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures on the symplectic cohomology of open symplectic manifolds. Using the SFT of Hamiltonian mapping tori we show how to define a homotopy extension of the well-known Lie bracket on symplectic cohomology. Apart from discussing applications to the existence of closed Reeb orbits, we outline how the $L_{\infty}$-structure is conjecturally related via mirror symmetry to the extended deformation theory of complex structures.Comment: Results of arXiv:1310.6014 got merged into arXiv:1412.2682, now entitled "Higher algebraic structures in Hamiltonian Floer theory" and published in Advances in Geometry (DOI: 10.1515/advgeom-2019-0017). Extensions of other announced results have been turned into an ongoing PhD thesis projec

On the Maslov class rigidity for coisotropic submanifolds

We define the Maslov index of a loop tangent to the characteristic foliation of a coisotropic submanifold as the mean Conley--Zehnder index of a path in the group of linear symplectic transformations, incorporating the "rotation" of the tangent space of the leaf -- this is the standard Lagrangian counterpart -- and the holonomy of the characteristic foliation. Furthermore, we show that, with this definition, the Maslov class rigidity extends to the class of the so-called stable coisotropic submanifolds including Lagrangian tori and stable hypersurfaces.Comment: 18 pages; v2 minor corrections, references update