Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry

Abstract

We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.

v2: slightly expanded exposition

References in corpus (4)

• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Infinity inner products on A-infinity algebras
• Sheaves on local Calabi-Yau varieties
• Homological mirror symmetry for Brieskorn-Pham singularities

Cited by in corpus (22)

• The monotone wrapped Fukaya category and the open-closed string map
• On the Fukaya category of a Fano hypersurface in projective space
• Symplectic cohomology and q-intersection numbers
• Integrable hierarchies and the mirror model of local CP1
• Lagrangian torus fibrations and homological mirror symmetry for the conifold
• Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures
• Homological mirror symmetry for the quintic 3-fold
• Lagrangian homology spheres in (A_m) Milnor fibres via C^*-equivariant A_infinity modules
• A Log PSS morphism with applications to Lagrangian embeddings
• Dual torus fibrations and homological mirror symmetry for A_n-singularities
• Derived equivalence classification of Brauer graph algebras
• A proof of N.Takahashi's conjecture for $({\mathbb{P}}^2,E)$ and a refined sheaves/Gromov-Witten correspondence
• Mirror symmetry and Fukaya categories of singular hypersurfaces
• Remarks on quiver gauge theories from open topological string theory
• Homological mirror symmetry for Brieskorn-Pham singularities
• Koszul duality via suspending Lefschetz fibrations
• Exact Lefschetz fibrations associated with dimer models
• Floer cohomology and pencils of quadrics
• A Double Poisson Algebra Structure on Fukaya Categories
• A note on homological mirror symmetry for singularities of type D
• Remarks on mirror symmetry of Donaldson-Thomas theory for Calabi-Yau 4-folds
• The Physical Mirror Equivalence for the Local P^2