On the symplectic cohomology of log Calabi-Yau surfaces

Abstract

This article studies the symplectic cohomology of affine algebraic surfaces that admit a compactification by a normal crossings anticanonical divisor. Using a toroidal structure near the compactification divisor, we describe the complex computing symplectic cohomology, and compute enough differentials to identify a basis for the degree-zero part of the symplectic cohomology. This basis is indexed by integral points in a certain integral affine manifold, providing a relationship to the theta functions of Gross--Hacking--Keel. Included is a discussion of wrapped Floer cohomology of Lagrangian submanifolds and a description of the product structure in a special case. We also show that, after enhancing the coefficient ring, the degree--zero symplectic cohomology defines a family degenerating to a singular surface obtained by gluing together several affine planes.

arXiv comment61 pages, 8 figures; v3 is a fairly significant revision in response to a referee's comments, with most of the changes occurring in section 6. Other remarks added and minor changes throughout. To appear in Geometry and Topology

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• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• String topology with gravitational descendants, and periods of Landau-Ginzburg potentials
• Lagrangian sections on mirrors of toric Calabi-Yau 3-folds
• The Higher Dimensional Tropical Vertex
• On the embedding complexity of Liouville manifolds
• Intrinsic mirror symmetry and categorical crepant resolutions
• Mirror Symmetry for Truncated Cluster Varieties
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• Koszul duality via suspending Lefschetz fibrations
• On the complement of a positive normal crossing divisor with no triple intersection in a projective variety
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• Geometry of symplectic log Calabi-Yau pairs
• Mirror symmetry for the Tate curve via tropical and log corals
• Circular spherical divisors and their contact topology
• The wrapped Fukaya category for semi-toric SYZ fibrations
• Maurer-Cartan deformation of Lagrangians