A categorification of Morelli's theorem

Bohan Fang, Chiu Chu Melissa Liu, David Treumann, Eric Zaslow

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

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 = Lie(T ≅ ℝn be the Lie algebra of the compact dual (real) torus T ≅ U(1)n. Then there is a corresponding conical Lagrangian Λ⊂T*M and an equivalence of triangulated dg categories PerfT(X) ≅ Shcc(M; ∨), where PerfT(X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Shcc(M;Λ) is the triangulated dg category of complex of sheaves on M 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 M.

Original languageEnglish (US)
Pages (from-to)79-114
Number of pages36
JournalInventiones Mathematicae
Volume186
Issue number1
DOIs
StatePublished - Oct 1 2011

ASJC Scopus subject areas

  • Mathematics(all)

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