A geometric criterion for generating the Fukaya category

By M. Abouzaid. 2010

196 citations recorded by the source corpus.

Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the identity in symplectic cohomology lies in the image of this map, we conclude that every Lagrangian lies in the idempotent closure of the chosen collection. The main new ingredients are (1) the construction of operations on the Fukaya category controlled by discs with two outputs, and (2) the Cardy relation.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/19cb81885e369cd488b59bf6479ba1e49c915924

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Legendrian fronts for affine varieties (2016, methodology): “- Intersections of the curve which do not have a thick dot are genuine transverse intersections of the Legendrian front, and in this case the Legendrian is extended in transverse directions by a rotation whose axis does not pass through the intersection.”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “A noncommutative divisor on A consists of an A∞-bimodule P which is invertible (with respect to tensor product of bimodules, up to quasi-isomorphism), together with an A∞-algebra structure on B = A ⊕ P[1], which extends the given one on A as well as the A-bimodule structure of P.”
Infinitely many Lagrangian fillings (2020, background): “The wrapped Fukaya categories of the Weinstein manifolds M(n,m) are generated by their respective unique cocore C of their defining 2-handle [1, 15], i.”
Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture (2011, background): “1 of [3], which asserts that the Fukaya category of a symplectic manifold is split-generated by a Lagrangian L whenever a certain geometric criterion involving the Hochschild homology of the Floer complex CF ∗(L,L) is satisfied.”
Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors (2017, methodology): “The above result implies in particular that Abouzaid’s generation criterion [2] is satisfied for the subcategory consisting of the cocore planes of a Weinstein manifold, from which one can conclude that the cocores split-generate the wrapped Fukaya category.”
The monotone wrapped Fukaya category and the open-closed string map (2012, methodology): “Abouzaid [2] proved the generation criterion in the exact setup using quadratic Hamiltonians, and Ganatra uses the same setup in his discussion [25] of the module structure of the string maps.”
Mirror symmetry: from categories to curve counts (2015, methodology): “We call any such A an essential sub-category ; the work of [Abo10], implemented for relative Fukaya categories in [ PS], implies that any such A split-generates F ( X ).”
Towards Mirror Symmetry for Varieties of General Type (2012, unknown): “ Fuk(S) and the middle one is the quantum cohomology of S. These are conjectured to be isomorphisms under certain conditions, see [Ko94], [AFOO] and for references with SH in place of QH see [Sei07], [Ab10], [Ga]. For the following considerations, let us assume that αis an isomorphism. On the comple”
Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, background): “Proof A quasi-isomorphism is given by X[1] Y Y[1] Z X[1] Z −α[1] −1[1] −β[1] −µ2(β, α)[1] −β 11[1] with quasi-inverse X[1] Y Y[1] Z X[1] Z −α[1] −1[1] −β[1] −µ2(β, α)[1] 11[1] −α[1] Definition 3.1.2 Given a morphism a ∈ hom0(X[s],X), we set a1 := a and define ak ∈ hom0(X[ks],X), k ≥ 1, inductively b”
Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (2010, background): “The main result of [5] is that this condition implies that every exact Lagrangian in M which, away from a compact set, is a cone on a Legendrian manifold of the ideal contact boundary of M , can be obtained from L by taking cones and summands; i.”
Floer cohomology and pencils of quadrics (2010, methodology): “Abouzaid [1] proves (in the exact case, and for wrapped Floer cohomology) that if the unit is in the image of the first map, then A split-generates the Fukaya category.”
The family Floer functor is faithful (2014, methodology): “Since the construction of the homotopy from the composition of the two maps we construct to the identity uses a moduli space of annuli, faithfulness can be seen as the analogue of the generation criterion [1].”