Altering symplectic manifolds by homologous recombination

Abstract

We use symplectic cohomology to study the non-uniqueness of symplectic structures on the smooth manifolds underlying affine varieties. Starting with a Lefschetz fibration on such a variety and a finite set of primes, the main new tool is a method, which we call homologous recombination, for constructing a Lefschetz fibration whose total space is smoothly equivalent to the original variety, but for which symplectic cohomology with coefficients in the given set of primes vanishes (there is also a simpler version that kills symplectic cohomology completely). Rather than relying on a geometric analysis of periodic orbits of a flow, the computation of symplectic cohomology depends on describing the Fukaya category associated to the new fibration. As a consequence of this and a result of McLean we prove, for example, that an affine variety of real dimension greater than 4 supports infinitely many different (Wein)stein structures of finite type, and, assuming a mild cohomological condition, uncountably many different ones of infinite type. In addition, we introduce a notion of complexity which measures the number of handle attachments required to construct a given Weinstein manifold, and prove that, in dimensions greater than or equal to 12, one may ensure that the infinitely many different Weinstein manifolds smoothly equivalent to a given algebraic variety have bounded complexity.

35 pages, 6 figures. Expanded discussion of the construction of symplectic cohomology in the SFT framework with arbitrary coefficients

References in corpus (4)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• An exact sequence for contact- and symplectic homology
• Effect of Legendrian Surgery
• A geometric criterion for generating the Fukaya category

Cited by in corpus (18)

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• A stable infinity-category of Lagrangian cobordisms
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• Effect of Legendrian Surgery
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• Loose Legendrians and the plastikstufe
• Symplectic fillings of asymptotically dynamically convex manifolds I
• Floer Cohomology, Multiplicity and the Log Canonical Threshold
• Distinguishing between exotic symplectic structures
• Prime-localized Weinstein subdomains
• On the embedding complexity of Liouville manifolds
• On the minimal symplectic area of Lagrangians
• Lefschetz fibrations on cotangent bundles and some plumbings
• Localization and flexibilization in symplectic geometry
• Exotic families of symplectic manifolds with Milnor fibers of $ADE$-type
• On the Theory of Stein Manifolds
• Non-standard Symplectic Structures via Symplectic Cohomology
• The infinity-category of stabilized Liouville sectors