The Fukaya category of symplectic neighborhood of a non-Hausdorff manifold
نویسنده
چکیده
§1.1 Background: Symplectic geometry (topology) is one of the fastest growing branch of geometry (topology) that exhibits both the characteristics of geometry and topology. Most of the new developments start with the well known fact that for any symplectic manifold (M,ω), there exists a compatible almost complex structure J on M (usually not integrable) that gives rise to the natural almost Kähler structure (M,J, g, ω). In the following we will discuss 3 such developments relevant to us that profoundly altered the landscape of symplectic geometry.
منابع مشابه
Fukaya Categories as Categorical Morse Homology
The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to t...
متن کاملA Symplectic Prolegomenon
A symplectic manifold gives rise to a triangulated A∞-category, the derived Fukaya category, which encodes information on Lagrangian submanifolds and dynamics as probed by Floer cohomology. This survey aims to give some insight into what the Fukaya category is, where it comes from, and what symplectic topologists want to do with it. ...everything you wanted to say required a context. If you gav...
متن کاملSeidel’s Mirror Map for the Torus
Paul Seidel had the following idea for recovering the mirror map purely from the Fukaya category.1 Start with a symplectic Calabi-Yau X and its family of complex structures, and assume it has a projective mirror manifold Y with a family of symplectic structures, and that Kontsevich’s conjecture holds: DFuk(X) ∼= D(Y ), where DFuk(X) is the Fukaya category ofX (i.e. the bounded derived category ...
متن کاملFukaya Algebras and the Minimal Model Program
We prove that small blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. We give examples of explicit mmp runnings and descriptions of Floer non-trivial tori in the case of toric manifolds, polygon spaces, and moduli spaces of flat bundles on punctured two-spheres (moduli of parabolic ...
متن کاملFukaya Categories and the Minimal Model Program: Creation
We prove that small blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with trivial centers create Floer-non-trivial Lagrangian tori. We give examples of explicit mmp runnings and descriptions of Floer non-trivial tori in the case of toric manifolds, polygon spaces, and moduli spaces of flat bundles on punctured two-spheres (moduli of paraboli...
متن کامل