Homogeneous coordinate rings and mirror symmetry for toric varieties

Abstract

Given a smooth toric variety $X$ and an ample line bundle $\mathcal{O}\left(1\right)$, we construct a sequence of Lagrangian submanifolds of ${\left({ℂ}^{\star }\right)}^n$ with boundary on a level set of the Landau–Ginzburg mirror of $X$. The corresponding Floer homology groups form a graded algebra under the cup product which is canonically isomorphic to the homogeneous coordinate ring of $X$.