  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.
