Homological mirror symmetry for curves of higher genus
type: Item
instance of: scholarly article
title: Homological mirror symmetry for curves of higher genus
zbMATH Open document ID: 1242.14039
author: Alexander I. Efimov
published in: Advances in Mathematics
publication date: 2012-06-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.3903
reviewed by: Pawel Sosna
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 53D37
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6040338
zbMATH Keywords: homological mirror symmetry
zbMATH Keywords: Fukaya categories
zbMATH Keywords: triangulated categories of singularities
zbMATH Keywords: formality theorem
MaRDI profile type: Publication
OpenAlex ID: W2015009262
arXiv ID: 0907.3903
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DOI: 10.1016/J.AIM.2012.02.022
review text: Kontsevich's homological mirror symmetry conjecture interprets, roughly speaking, mirror symmetry of two Calabi-Yau varieties \(X\) and \(X'\) as an equivalence of categories, namely the bounded derived category of coherent sheaves on \(X\) and the Fukaya category of \(X'\). The conjecture has been established in some cases, for example, for elliptic curves and the quartic \(K3\) surface. Later the conjecture was extended to Fano varieties and varieties of general type. In these cases the mirror is a Landau-Ginzburg model, that is, a smooth algebraic variety together with a regular function. \textit{P. Seidel} [J.\ Algebr.\ Geom.\ 20, No.\ 4, 727--769 (2011; Zbl 1226.14028)] proved homological mirror symmetry for genus 2 curves. The aim of the paper under review is to prove it in the case of curves of genus \(g\geq 3\).NEWLINENEWLINEThe curves are treated as symplectic varieties. So, let \(M\) be a symplectic compact oriented surface of genus \(\geq 3\). One considers \(\mathcal{F}(M)\), the Fukaya \(A_\infty\)-category of \(M\), and \(D^\pi(\mathcal{F}(M))\) the category of perfect complexes over \(\mathcal{F}(M)\). On the algebraic side, the Landau--Ginzburg model \(W: X\rightarrow \mathbb{C}\) is three-dimensional. It has one singular fibre \(H\), which is a union of \((g+1)\) complex surfaces. The category of singularities of \(H\), that is, the Verdier quotient of the bounded derived category of coherent sheaves on \(H\) by the triangulated subcategory of perfect complexes, is denoted by \(D_{sg}(H)\). The main result of the paper states that the triangulated categories \(D^\pi(\mathcal{F}(M))\) and \(\overline{D_{sg}}(H)\), where the latter is the Karoubian completion of \(D_{sg}(H)\), are equivalent.NEWLINENEWLINEThe basic idea of the proof is the same as in Seidel's paper. Both categories can be described by \(A_\infty\)-algebras of a very special form, namely \(A_\infty\)-deformations of the exterior algebra. On the algebraic side, the structure sheaf of th…
description: scientific article; zbMATH DE number 6040338