On the homological mirror symmetry conjecture for pairs of pants (Q431565)

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.