Altering symplectic manifolds by homologous recombination

TL;DR

This paper introduces homologous recombination, a new method to alter symplectic structures on affine varieties, demonstrating the existence of infinitely many distinct Weinstein structures with controlled complexity.

Contribution

The authors develop homologous recombination to construct Lefschetz fibrations with vanishing symplectic cohomology, revealing multiple Weinstein structures on the same underlying manifold.

Findings

Affine varieties of dimension >4 support infinitely many Weinstein structures.

Uncountably many Weinstein structures of infinite type exist under mild conditions.

In dimensions ≥12, these structures can have bounded complexity.

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.