Volume 23, issue 6 (2019)

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On the symplectic cohomology of log Calabi–Yau surfaces

James Pascaleff

Geometry & Topology 23 (2019) 2701–2792
Abstract

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.

Keywords
symplectic cohomology, log Calabi–Yau surface, affine manifold, wrapped Floer cohomology
Mathematical Subject Classification 2010
Primary: 53D40
Secondary: 14J33, 53D37
References
Publication
Received: 14 August 2013
Revised: 2 December 2018
Accepted: 7 March 2019
Published: 1 December 2019
Proposed: Yasha Eliashberg
Seconded: Peter Ozsváth, Jim Bryan
Authors
James Pascaleff
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