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

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Abstract

The coherent-constructible (CC) correspondence is a relationship between coherent sheaves on a toric variety X and constructible sheaves on a real torus \(\mathbb {T}\). This was discovered by Bondal and established in the equivariant setting by Fang, Liu, Treumann, and Zaslow. In this paper, we explore various aspects of the non-equivariant CC correspondence. Also, we use the non-equivariant CC correspondence to prove the existence of tilting complexes in the derived categories of toric orbifolds satisfying certain combinatorial conditions. This has applications to a conjecture of King.

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Notes

  1. A different but closely related result is due to Kawamata [27] who proves that all toric varieties admit exceptional collections composed of coherent sheaves.

  2. A related statement is proposed as Conjecture 1.6 in [16].

  3. For an introduction to mirror symmetry with special emphasis on Kontsevich’s HMS, we recommend the nice overview [5].

  4. Bondal’s original statement involved considering first a stratification \(\mathcal {S}_\mathbf {\Sigma }\) on \(\mathbb {T}\) capturing the combinatorics of \(\mathbf {\Sigma }\), and then the subcategory of sheaves constructible with respect to \(\mathcal {S}_\mathbf {\Sigma }\), \( Sh _c(\mathbb {T}, \mathcal {S}_\mathbf {\Sigma })\). However, the categories \( Sh _c(\mathbb {T}, \mathcal {S}_\mathbf {\Sigma })\) and \( Sh _c(\mathbb {T}, \varLambda _\mathbf {\Sigma })\) coincide: This follows from a mild generalization to non-Whitney stratifications of Remark 4.5 below.

  5. We stress that here we are working with the ordinary tensor product of constructible sheaves, as opposed to the convolution product which was discussed in Sect. 5.1.

  6. Note that \(p^! {\mathcal {F}} \in Sh _c(M, {\tilde{\varLambda }}_{\mathbf {\Sigma }})\) is not compactly supported: However, the tensor product \({\tilde{\kappa }}({\tilde{\mathcal {L}}}_j) \otimes p^! {\mathcal {F}}\) is, and therefore lies in \( Sh _{cc}(M, {\tilde{\varLambda }}_{\mathbf {\Sigma }})\) as required.

  7. All algebras appearing in this paper are always assumed to be associative and unital.

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Acknowledgments

We thank David Treumann and Eric Zaslow for their interest in this project. We thank particularly David Treumann for many useful discussions. Claus Ringel kindly answered some of our questions. S.S. thanks the Mathematical Institute for funding and providing a stimulating environment. N.S. thanks the Max Planck Institute for Mathematics for excellent working conditions.

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  1. Mathematical Institute of the University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Sarah Scherotzke

  2. Mathematics Department, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada

    Nicolò Sibilla

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  1. Sarah Scherotzke
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  2. Nicolò Sibilla
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Correspondence to Nicolò Sibilla.

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Scherotzke, S., Sibilla, N. The non-equivariant coherent-constructible correspondence and a conjecture of King. Sel. Math. New Ser. 22, 389–416 (2016). https://doi.org/10.1007/s00029-015-0193-y

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