Homological mirror symmetry for curves of higher genus
triangulated categories of singularities
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 the origin is a split-generator in the category of singularities and the wanted algebra is its endomorphism DG-algebra. On the symplectic side, one considers a cyclic covering \(M\rightarrow \overline{M}\), where \(\overline{M}\) is \(\mathbb{P}^1\) with three orbifold points. The wanted split-generator is then the direct sum of \(2g+1\) curves in \(M\), which form a Galois-invariant collection and project to the same curve in \(\overline{M}\). The wanted algebra is again the endomorphism algebra of the generator.NEWLINENEWLINEIn addition to the above result, the author also proves a kind of reconstruction theorem for hypersurface singularities.
scientific article; zbMATH DE number 6040338
scientific article