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.

61 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

References in corpus (2)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Symplectic cohomology and duality for the wrapped Fukaya category

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