The nonequivariant coherent-constructible correspondence for toric surfaces

Abstract

Abstract. We prove the nonequivariant coherent-constructible correspondence conjectured by Fang-Liu-Treumann-Zaslow in the case of toric surfaces. Our proof is based on describing a semiorthogonal decomposition of the constructible side under toric point blow-up and comparing it with Orlov's theorem.

Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

“…It should be noted that in the microlocal homological mirror symmetry for toric varieties introduced by Fang-Liu-Treumann-Zaslow in [19,20,21,22] and proved in full generality by Kuwagaki in [39], there is also an induced monoidal equivalence with respect to a monoidal structure on the category of microlocal sheaves. Also, the monoidal structure that we defined above fits well with the work of Subotic in [55] where a monoidal structure for Fukaya categories is described as fiberwise addition in the presence of a Lagrangian torus fibration with a reference section (in our case, these are the moment map µ and the section L, respectively).…”

“…In a different direction, it is shown in [20] that there is a derived equivalence between the category of torus equivariant coherent sheaves on X and the category of constructible sheaves on R n , which should be viewed as the universal cover the base torus of T * T n ≃ (C * ) n , with microsupport in a Lagrangian skeleton L Σ determined by Σ. The nonequivariant case is shown in certain cases in [38,46,57] and has recently been shown to hold in great generality (without any Fano assumptions and in the singular case with coherent sheaves replaced by perfect complexes) in [39]. Categories of microlocal sheaves were first related to infinitesimally wrapped Fukaya category by [42] and [43].…”

Kasteleyn operators from mirror symmetry

“…Associated to Σ is a toric DM stack X Σ which has no nontrivial stabilizers on the open subspace T N ⊂ X Σ and whose coarse moduli space is X Σ (see e.g. [Kuw16,§5]). When β is an isomorphism X Σ has no nontrivial stabilizers at all and coincides with the variety X Σ .…”

“…We begin by recalling that since π is proper the coherent-constructible correspondence intertwines the pullback π * : Coh(X Σ • ) ֒→ Coh(X Σ • ) with the trivial inclusion id * T 2 : Sh w Λ Σ • (T 2 ) ֒→ Sh w Λ Σ • (T 2 ) (by [Kuw16,Prop. 9.3] in the present generality, following [FLTZ11, Tre10, SS14]).…”

“…between the dg categories of perfect complexes and constructible sheaves is referred to as the coherent-constructible correspondence (or CCC) [FLTZ11], proved in the stated generality in [Kuw16]. Local systems on the graph Γ give rise to Lagrangian branes in T * T 2 via the construction of [STWZ15]: up to Hamiltonian isotopy there is a canonical embedded exact Lagrangian L Γ in T * T 2 which deformation retracts onto Γ.…”