Lagrangian tori in four-dimensional Milnor fibres
type: Item
instance of: scholarly article
title: Lagrangian tori in four-dimensional Milnor fibres
zbMATH Open document ID: 1338.32025
author: Ailsa M. Keating
published in: Geometric and Functional Analysis. GAFA
publication date: 2016-01-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1405.0744
review text: This paper constructs exact Lagrangian tori in the Milnor fibres of all non-simple hypersurface singularities of dimension two. It suffices to do this for the \(T_{p,q,r}\)-singularities, as the Milnor fibre of an adjacent singularity can be symplectically embedded in the original Milnor fibre. The basis of the construction is A'Campo's explicit model for the Milnor fibre of a plane curve singularity, starting from a real divide. By suspension the result for \(T_{p,q,2}\) follows directly, while for \(r\geq 3\) a deformation of suitable suspension is used. The explicit model allows to compute Floer cohomologies, and this is used to show that the Fukaya category of the Milnor fibre of \(T_{p,q,r}\) is not split-generated by any collection of vanishing cycles. Also the bounded derived directed Fukaya category is computed, and it is shown that it is equivalent to the bounded derived category of coherent sheaves on an orbifold \(\mathbb P^1\) with \((1/p,1/q,1/r)\)-isotropy.
reviewed by: Jan Stevens
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 53D37
Mathematics Subject Classification ID: 32S55
zbMATH DE Number: 6526261
zbMATH Keywords: Milnor fibre
zbMATH Keywords: vanishing cyle
zbMATH Keywords: Lagrangian torus
zbMATH Keywords: unimodal singularities
zbMATH Keywords: Fukaya category
MaRDI profile type: Publication
OpenAlex ID: W2963593421
arXiv ID: 1405.0744
cites work: An open string analogue of Viterbo functoriality
cites work: Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I
cites work: Real deformations and complex topology of plane curve singularities
cites work: A Lagrangian Piunikhin-Salamon-Schwarz Morphism and Two Comparison Homomorphisms in Floer Homology
cites work: Dynamical systems I. Ordinary differential equations and smooth dynamical systems. Transl. from the Russian
cites work: Q5545282
cites work: Q4107251
cites work: Mirror symmetry and T-duality in the complement of an anticanonical divisor
cites work: Mirror symmetry for weighted projective planes and their noncommutative deformations
cites work: REPRESENTATION OF ASSOCIATIVE ALGEBRAS AND COHERENT SHEAVES
cites work: Homological mirror symmetry for A n -resolutions as a T -duality
cites work: Deformations of symplectic cohomology and exact Lagrangians in ALE spaces
cites work: Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for \(\mathbb{P}^1_{a,b,c}\)
cites work: Lagrangian Floer potential of orbifold spheres
cites work: Séminaire sur les singularités des surfaces. Centre de Mathématiques de l'École Polytechnique, Palaiseau 1976-1977
cites work: Singularities and topology of hypersurfaces
cites work: Fifteen characterizations of rational double points and simple critical points
cites work: Mirror Symmetry between Orbifold Curves and Cusp Singularities with Group Action
cites work: Symplectic mapping class groups of some Stein and rational surfaces
cites work: Homological mirror symmetry for Brieskorn-Pham singularities
cites work: Intersection matrices for certain singularities
cites work: Dynkin diagrams for unimodal singularities
cites work: Dynkin diagrams for singularities of functions of two variables
cites work: Fundamental groups of projective discriminant complements
cites work: Q5470192
cites work: Stability conditions on \(A_n\)-singularities
cites work: Complexifications of Morse functions and the directed Donaldson-Fukaya category
cites work: Spin Structures and Quadratic forms on Surfaces
cites work: Dehn twists and free subgroups of symplectic mapping class groups
cites work: Quivers, Floer cohomology, and braid group actions
cites work: The symplectic topology of some rational homology balls
cites work: The Invariance of Milnor's Number Implies the Invariance of the Topological Type
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: The surgery of Lagrange submanifolds
cites work: Un résultat sur la monodromie. (A result on the monodromy)
cites work: Lagrangian two-spheres can be symplectically knotted
cites work: Graded lagrangian submanifolds
cites work: Fukaya categories and Picard-Lefschetz theory
cites work: Homological Mirror Symmetry for the genus two curve
cites work: Fukaya \(A_\infty\)-structures associated to Lefschetz fibrations. I
cites work: Weighted Projective Lines Associated to Regular Systems of Weights of Dual Type
cites work: Special Lagrangians, stable bundles and mean curvature flow
cites work: THE TOPOLOGICAL PROPERTIES OF ISOLATED SINGULARITIES OF COMPLEX SPACES OF CODIMENSION ONE
cites work: Idéaux de fonctions différentiables. I
cites work: Dual torus fibrations and homological mirror symmetry for \(A_n\)-singularities
cites work: Exact Lagrangians in \(A_n\)-surface singularities
DOI: 10.1007/S00039-015-0353-4
description: scientific article