Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

By M. Abouzaid. 2006

141 citations recorded by the source corpus.

Abstract.Given a smooth projective toric variety X, we construct an A∞ category of Lagrangians with boundary on a level set of the Landau–Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3938b9ae1cf0813aa60dab8aefa7e875a1e9edd5

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On partially wrapped Fukaya categories (2016, background): “Mirror symmetry in various forms has been proved via this approach in [1, 17], and many others.”
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].”
Homological mirror symmetry for punctured spheres (2011, background): “⌋ k=1 k−1 2 Hk+ Xn k=⌊n 2 ⌋+1 n−k 2 −1 Hk. It is straightforward to check that for i≥ 0 the restriction of OX0(L− H1 − ··· − Hi) to Hi+1 is trivial. Moreover, the morphism ui,i+1 : Ei→ Ei+1[1] for i≥ 1 can be interpreted as follows. Let f: Ei∼= OH i(L−H1 −···−Hi−1) → OS j6= iH j (L−H1 −···−Hi)[”
Duality between Lagrangian and Legendrian invariants (2017, background): “⊗k sa1, where for a ∈ A , sa ∈ A [1] denotes the corresponding element in A [1] with degree shifted down by 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].”