On the homological mirror symmetry conjecture for pairs of pants
type: Item
instance of: scholarly article
title: On the homological mirror symmetry conjecture for pairs of pants
zbMATH Open document ID: 1255.53065
author: Nick Sheridan
published in: Journal of Differential Geometry
publication date: 2012-06-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.3238
full work available at URL: https://projecteuclid.org/euclid.jdg/1324477412
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 53D37
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 53D40
zbMATH DE Number: 6050961
zbMATH Keywords: mirror symmetry
zbMATH Keywords: Lagrangian sphere
zbMATH Keywords: Fukaya category
zbMATH Keywords: pair of pants
zbMATH Keywords: Fermat hypersurfaces
MaRDI profile type: Publication
OpenAlex ID: W2963747854
Wikidata QID: Q115170458
arXiv ID: 1012.3238
DOI: 10.4310/JDG/1324477412
review text: A pair of pants is a smooth complex affine variety given by \(\sum z_j=0\) in \(\mathbb{CP}^{n+2}-\{ \prod z_j=0\}\). The author constructs an immersed Lagrangian sphere \(L\) in the pair of pants and he computes the corresponding Floer cohomology algebra \(HF^*(L,L)\). Under the Homological Mirror Symmetry (HMS) the mirror of the pair of pants is conjectured to be the Landau--Ginzburg model \((\mathbb{C}^{n+2}, W=\prod z_j)\). In this case \(L\) should correspond to the structure sheaf of the origin in the category of singularities of \(W^{-1}(0)\) and one can check that we indeed have the conjectured isomorphism of cohomology algebras. In fact, the author proves that this isomorphism extends, as predicted by the HMS, to a quasi-isomophism of \(A_{\infty}\) algebras.NEWLINENEWLINEAs an application the author considers the HMS conjecture for Fermat hypersurfaces. More precisely, in [Homological mirror symmetry for the quartic surface, \url{arXiv:math/0310414}] \textit{P. Seidel} proved, on the symplectic side, Kontsevich's mirror conjecture for a quartic surface in \(\mathbb{P}^3\). The paper under review generalizes Seidel's arguments to the Fermat hypersurface in a projective space of arbitrary dimension and it gives a partial result: \(A_{\infty}\) embedding of certain categories which is conjectured to be an equivalence.
description: scientific article; zbMATH DE number 6050961