Altering symplectic manifolds by homologous recombination
By M. Abouzaid, P. Seidel. 2010
47 citations recorded by the source corpus.
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.
View Paper: https://www.semanticscholar.org/paper/0bdcfda6df51f141f25947f5b5136ea654b54d0d
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
- Alexander F. Ritter
- M. Abouzaid
- A. Oancea
- P. Seidel
- Kai Cieliebak
- I. Smith
- Frederick J. Bourgeois III
- Mark McLean
Citations in the corpus
- Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “In [45], it was pointed out that these categories always come with an additional datum: a distinguished natural transformation from the Serre functor to the identity functor (see [41, 8, 46, 3] for related theoretical developments, and [27, 28, 4] for applications).”
- The monotone wrapped Fukaya category and the open-closed string map (2012, background): “The wrapped Fukaya category W(E) of a symplectic manifold E with convex contact boundary, introduced in [19, 8], is of increasing importance in both symplectic topology (topology of Stein manifolds [21, 9], the nearby Lagrangian conjecture [5]) and Homological Mirror Symmetry [6, 33].”
- Symplectic homology and the Eilenberg-Steenrod axioms (2015, background): “flexible Stein domains [25] as well as certain non-flexible Stein domains [60, 2, 62, 64].”