| Summary: | A symplectic manifold is a smooth manifold M together with a choice of aclosed non-degenerate two-form. Recent years have seen the importance of associatingan A∞-category to M, called its Fukaya category, in helping to understandsymplectic properties of M and its Lagrangian submanifolds. One of the principlesof this construction is that automorphisms of the symplectic manifold shouldinduce autoequivalences of the derived Fukaya category, although precisely whatautoequivalences are thus obtained has been established in very few cases. Given a Lagrangian V ≅ CPn in a symplectic manifold (M,ω), there is anassociated symplectomorphism ∅v of M. In Part I, we defi ne the notion of aCPn-object in an A∞-category A, and use this to construct algebraically an A∞-functor Φv , which we prove induces an autoequivalence of the derived categoryDA. We conjecture that Φv corresponds to the action of ∅v and prove this inthe lowest dimension n = 1. We also give examples of symplectic manifolds forwhich this twist can be defi ned algebraically, but corresponds to no geometricautomorphism of the manifold itself: we call such autoequivalences exotic. Computations in Fukaya categories have also been useful in distinguishing certainsymplectic forms on exact symplectic manifolds from the 'standard' forms. In Part II, we investigate the uniqueness of so-called exotic structures on certainexact symplectic manifolds by looking at how their symplectic properties changeunder small nonexact deformations of the symplectic form. This allows us to distinguishbetween two exotic symplectic forms on T*S3∪2-handle, even though thestandard symplectic invariants such as their Fukaya category and their symplecticcohomology vanish. We also exhibit, for any n, an exact symplectic manifoldwith n distinct, exotic symplectic structures, which again cannot be distinguishedby symplectic cohomology or by the Fukaya category.
|
|---|
