Fukaya categories and deformations

Abstract

This is an informal (and mostly conjectural) discussion of some aspects of Fukaya categories. We start by looking at exact symplectic manifolds which are obtained from a closed Calabi-Yau by removing a hyperplane section. We look at the possible geometric significance of Hochschild cohomology in this situation, and how one can try to get from the Fukaya category of the exact manifold to that of the closed Calabi-Yau. Also included is a brief discussion of the role of Lefschetz pencils, and a bit of general deformation theory. To appear in the Proceedings of the Beijing ICM.

References in corpus (2)

• Functors and Computations in Floer homology with Applications Part II
• Infinity-Inner-Products on A-Infinity-Algebras

Cited by in corpus (45)

• Bordered Heegaard Floer homology: Invariance and pairing
• Covariantly functorial wrapped Floer theory on Liouville sectors
• Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space
• Bimodules in bordered Heegaard Floer homology
• Symplectic cohomology and duality for the wrapped Fukaya category
• Homological mirror symmetry for the quartic surface
• Homological Mirror Symmetry for manifolds of general type
• The monotone wrapped Fukaya category and the open-closed string map
• Symplectic cohomology and q-intersection numbers
• D-brane categories
• Fukaya categories of the torus and Dehn surgery
• Symplectic cohomology and Viterbo's theorem
• On the combinatorics of exact Lagrangian surfaces
• Floer theory and reduced cohomology on open manifolds
• The symplectic arc algebra is formal
• Arithmetic mirror symmetry for the 2-torus
• Topological conformal field theories and Calabi-Yau categories
• The Gromov-Witten potential associated to a TCFT
• From symplectic cohomology to Lagrangian enumerative geometry
• Homotopical and Higher Categorical Structures in Algebraic Geometry
• Mirror symmetry for log Calabi-Yau surfaces I
• Helix Structures in Quantum Cohomology of Fano Varieties
• Automatically generating Fukaya categories and computing quantum cohomology
• A biased view of symplectic cohomology
• Open strings, Lagrangian conductors and Floer functor
• Cyclic A_\infty Structures and Deligne's Conjecture
• Homological mirror symmetry for $A_n$-resolutions as a $T$-duality
• Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces
• XI Solomon Lefschetz Memorial Lecture Series: Hodge structures in non-commutative geometry. (Notes by Ernesto Lupercio)
• Special Lagrangian submanifolds of log Calabi-Yau manifolds
• Perverse sheaves of categories and some applications
• Slow entropy and symplectomorphisms of cotangent bundles
• Homological mirror symmetry at large volume
• Floer homology of Lagrangian submanifolds
• Homological mirror symmetry for punctured spheres
• Pairings in mirror symmetry between a symplectic manifold and a Landau-Ginzburg $B$-model
• Disjoinable Lagrangian tori and semisimple symplectic cohomology
• Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres
• Homological Mirror Symmetry for local SYZ singularities
• Versality in mirror symmetry
• Hyperplane sections and stable derived categories
• Curved String Topology and Tangential Fukaya Categories II
• Categorical non-properness in wrapped Floer theory
• Monotone Lagrangian Floer theory in smooth divisor complements: I
• Cyclic Homology of Fukaya Categories and the Linearized Contact Homology