Homogeneous coordinate rings and mirror symmetry for toric varieties

In this paper we give some evidence for M Kontsevich’s homological mirror symmetry conjecture [13] in the context of toric varieties. Recall that a smooth complete toric variety is given by a simplicial rational polyhedral fan ∆ such that |∆| = Rn and all maximal cones are non-singular (Fulton [10, Section 2.1]). The convex hull of the primitive vertices of the 1–cones of ∆ is a convex polytope which we denote by P, containing the origin as an interior point, and may be thought of as the Newton polytope of a Laurent polynomial W : (C?)n → C. This Laurent polynomial is the Landau–Ginzburg mirror of X .