On the symplectic cohomology of log Calabi–Yau surfaces

Pascaleff, James¹

¹ Department of Mathematics, University of Texas at Austin, Austin, TX, United States, Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, United States

Geometry & topology, Tome 23 (2019) no. 6, pp. 2701-2792.

Résumé

We study 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 and 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.

Détail

DOI : 10.2140/gt.2019.23.2701

Classification : 53D40, 14J33, 53D37
Keywords: symplectic cohomology, log Calabi–Yau surface, affine manifold, wrapped Floer cohomology

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