Altering symplectic manifolds by homologous recombination

Details Altering symplectic manifolds by homologous recombination Altering symplectic manifolds by homologous recombination

2010-07-19

arxiv.org/abs/1007.3281v3

Mathematics

Symplectic Geometry

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

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Altering symplectic manifolds by homologous recombination does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Altering symplectic manifolds by homologous recombination, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Altering symplectic manifolds by homologous recombination will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-124618