The non-equivariant coherent-constructible correspondence and a conjecture of King

Abstract

Abstract. The coherent-constructible correspondence is a relationship between coherent sheaves on a toric variety X and constructible sheaves on a real torus T. This was discovered by Bondal, and explored in the equivariant setting by Fang, Liu, Treumann and Zaslow. In this paper we prove the equivariant coherent-constructible correspondence for a class of toric varieties including weighted projective space. Also, we give applications to the construction of tilting complexes in the derived category of toric DM stacks.

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

“…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].…”

Categorical Localization for the Coherent-Constructible Correspondence

“…Theorem 2.10 ([Tre10, SS16,Kuw17b]). If Σ is cragged or dim Σ = 2, the morphism κ Σ is a quasi-equivalence.…”

“…Conjecture 1.1 is proved in some special cases ([Tre10, SS16,Kuw17b]) and in the equivariant version ([FLTZ11]). Conjecture 1.2 is sketched in [Vai] and also proved in [Kuw17a] independently.…”

The nonequivariant coherent-constructible correspondence for toric surfaces

“…On the coherent side, we will use the following four stable ∞-categories: FLTZ12,Tre10]. Scherotzke-Sibilla [SS16] generalized the conjecture for complete oribfolds, and Ike and the author [IK16] generalized further for smooth fans. For complete fans, the conjecture was proved when Σ is (i) zonotpal by Treumann [Tre10], (ii) cragged by Scherotzke-Sibilla [SS16], and (iii) 2-dimensional by the present author [Kuw15].…”

“…There exists an equivalence of ∞-categories There may also be potential applications of Theorem 1.2 to problems in the derived category of coherent sheaves on toric varieties. In fact, Fang-Liu-Treumann-Zaslow applied their equivariant version to prove Kawamata's semi-orthogonal decomposition [FLTZ11b] and Scherotzke-Sibilla applied to construction of tilting complexes in the derived categories of coherent sheaves of cragged toric stacks [SS16].…”