A categorification of Morelli’s theorem

By Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, Eric Zaslow. 2010

85 citations recorded by the source corpus.

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli’s description of the K-theory of a smooth projective toric variety (Morelli in Adv. Math. 100(2):154–182, 1993). Specifically, let X be a proper toric variety of dimension n and let $M_{\mathbb{R}} = \mathrm{Lie}(T_{\mathbb{R}}^{\vee})\cong\mathbb {R}^{n}$ be the Lie algebra of the compact dual (real) torus $T_{\mathbb{R}}^{\vee}\cong U(1)^{n}$. Then there is a corresponding conical Lagrangian Λ⊂T∗Mℝ and an equivalence of triangulated dg categories $\mathcal{P}\mathrm{erf}_{T}(X) \cong\mathit{Sh}_{cc}(M_{\mathbb{R}};\Lambda)$, where $\mathcal{P}\mathrm{erf}_{T}(X)$ is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Shcc(Mℝ;Λ) is the triangulated dg category of complex of sheaves on Mℝ with compactly supported, constructible cohomology whose singular support lies in Λ. This equivalence is monoidal—it intertwines the tensor product of coherent sheaves on X with the convolution product of constructible sheaves on Mℝ.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0726bd2c05f37ab1c7b63c1a38e3c4ee0b399ed3

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks (2009, methodology): “The result of [19] says the category of equivariant coherent sheaves on P(1, 1, 2) is quasi-equivalent to a subcategory of constructible sheaves on the”
On partially wrapped Fukaya categories (2016, methodology): “The equivalence proceeds by combining the coherent-constructible correspondence [13] and the Nadler–Zaslow correspondence [29].”
T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, methodology): “In this subsection, we briefly recall the results of [14,33].”