Twisted polytope sheaves and coherent–constructible correspondence for toric varieties

Abstract

Given a smooth projective toric variety X Σ of complex dimension n, Fang-Liu-Treumann-Zaslow [FLTZ] showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves Coh(X Σ ) into the dg derived category of constructible sheaves on a torus Sh(T n , Λ Σ ). Recently, Kuwagaki [Ku2] proved that the quasi-embedding is a quasi-equivalence, and generalized the result to toric stacks. Here we give a different proof in the smooth projective case, using non-characteristic deformation of sheaves to find twisted polytope sheaves that co-represent the stalk functors. arXiv:1701.00689v1 [math.AG] 3 Jan 2017 1 One needs to be careful about the endpoint t = 1, since SS ∞ (P 1 ) ∩ Λ ∞ Σ = ∅. The non-characteristic deformation lemma for sections over open sets, Proposition 2.7, avoids this problem.

Local mirror symmetry via SYZ

“…where in the first factor we write σ ⊥ for the image of the subspace orthogonal to σ under the projection R n+m → T n+m . This Lagrangian, studied in [8,7,9], following earlier work [5], is known [13,32] to be the skeleton of the Liouville-sectorial mirror to the toric variety 19,31,29], and its boundary 23,13].…”

Mirror symmetry for Berglund-Hübsch Milnor fibers

“…The Lagrangian L Σ was first studied by Bondal in [Bon06], then later studied extensively by Fang-Liu-Treumann-Zaslow [FLTZ12a,FLTZ11,FLTZ12b], for the relation between the category Sh −L Σ (N S 1 ) of constructible sheaves on N S 1 microsupported along −L Σ and the category QCoh(T Σ ) of quasi-coherent sheaves on the toric stack T Σ with fan Σ. Following the above works, a complete statement was first obtained in [Kuw20] (followed by another proof in many cases in [Zho19]). In modern language, the best statement reads as follows:…”

Mirror Symmetry for Truncated Cluster Varieties

“…The motivation for studying the Lagrangian L Σ was the following result, studied by Bondal and FLTZ and eventually proved in full generality by Kuwagaki (later reproved in many cases in [33]):…”