Homogeneous coordinate rings and mirror symmetry for toric varieties

Abstract

Given a smooth toric variety X and an ample line bundle O(1), we construct a sequence of Lagrangian submanifolds of (C^*)^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.Comment: This is the version published by Geometry & Topology on 24 August 200

Homological mirror symmetry is T-duality for $\mathbb{P}^n$

“…which is obviously the graph of the (S 1 ) n -valued function γ (1) . For any other holomorphic line bundle O(k), let h k denote (h 1 ) k , and this construction gives rise to a Lagrangian ( Figure 1).…”

“…Following this line, Auroux et al [5] prove homological mirror symmetry for weighted projective planes (and their non-commutative deformations) and Del Pezzo surfaces [6]. Abouzaid [1,2] proves the case of all smooth projective toric varieties using tropical geometry. Bondal and Ruan also announce a result for weighted projective spaces [7].…”

Seidel's mirror map for abelian varieties

“…Also note that using the (commutative) Fukaya product, we can express Z k in terms of X 3 , XY and Z. 1 Similarly, the results of [14] allow us to readily express the six points…”

“…(0,0) , X 1 := Y [1] (1,0) , X 2 := Y [1] (0,1) , X 3 := Y [1] (1,1) , we have X 0 X 3 = A [4] 1 A [4] 1 Y [2] (1,1) + Y [2] (1,3) + Y [2] (3,1) + Y [2] (3,3) = X 1 X 2 (5.1)…”

“…Hom DFuk(X) (ψ(O), ψ(O(k))), (1.1) where ψ is the equivalence of categories (see also [1]). In fact, in the above, O can be replaced by any line bundle L. As the mirror of the automorphism -⊗O Y (1) is known to be the symplectomorphism ρ effecting the monodromy around the large complex structure limit, the right-hand side can be computed entirely from the Fukaya category, once we choose any Fukaya object mirror to some line bundle.…”