Homogeneous coordinate rings and mirror symmetry for toric varieties

By M. Abouzaid. 2005

99 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7dee8f81622d16f0423c755b106c1e1b96a3ae17

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Morse homology, tropical geometry, and homological mirror symmetry for toric varieties (2006, methodology): “The open sets Oτ therefore correspond to Oτ̌ in the notation of [1].”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “Complete or partial results in the Fano case are known for toric varieties [13, 14, 15], del Pezzo surfaces [16], and weighted projective planes [17].”
Lagrangian cobordism. I (2011, background): “instance the works of Seidel [Sei3], Abouzaid [Abo], Auroux [Aur], as well as older work of Oh [Oh4]).”
Fukaya categories of symmetric products and bordered Heegaard-Floer homology (2010, result): “The description we give here is slightly different from that in Seidel’s work, but can easily be checked to be equivalent; it is also closely related to the viewpoint given by Abouzaid in Section 2 of [1], except we place the base point at infinity.”
T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, methodology): “2) and much closer to Abouzaid’s work [1,2].”
An open string analogue of Viterbo functoriality (2007, background): “However, for the authors of this paper, the main interest lies in potential applications to homological mirror symmetry, using Mikhalkin’s generalized pair-of-pants decompositions [2, 24].”
Mirror symmetry and T-duality in the complement of an anticanonical divisor (2007, background): “The formulation which is the most relevant to us is the one which appears in Abouzaid’s work [1, 2]: in this version, one considers Lagrangian submanifolds of M with boundary contained in a given fiber of the superpotential, and which near the reference fiber are mapped by W to an embedded curve γ ⊂”
On the homological mirror symmetry conjecture for pairs of pants (2010, background): “Complete or partial results in the Fano case are known for toric varieties [1, 2, 15], del Pezzo surfaces [8], and weighted projective planes [9].”
Mirror symmetry for toric Fano manifolds via SYZ transformations (2008, background): “, m, h(Ek) = h(V) + R≥0v(Ek) for some v(Ek) ∈ Σ[1], and the following balancing condition is satisfied:”