On the symplectic cohomology of log Calabi-Yau surfaces

TL;DR

This paper investigates the symplectic cohomology of certain affine surfaces with normal crossings divisors, linking it to affine geometry and theta functions, and explores its algebraic structure and degenerations.

Contribution

It provides a detailed description of the symplectic cohomology for log Calabi-Yau surfaces, connecting it to integral affine manifolds and theta functions, and analyzes its algebraic properties.

Findings

Basis for degree-zero symplectic cohomology indexed by integral points

Relationship established between symplectic cohomology and theta functions

Degeneration of symplectic cohomology to a singular surface

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.