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A categorification of Morelli's theorem
- Source :
- Invent. Math. 186 (2011), no.1, 79-114
- Publication Year :
- 2010
-
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. Specifically, let $X$ be a proper toric variety of dimension $n$ and let $M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n$ be the Lie algebra of the compact dual (real) torus $T_\bR^\vee\cong U(1)^n$. Then there is a corresponding conical Lagrangian $\Lambda \subset T^*M_\bR$ and an equivalence of triangulated dg categories $\Perf_T(X) \cong \Sh_{cc}(M_\bR;\Lambda),$ where $\Perf_T(X)$ is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on $X$ and $\Sh_{cc}(M_\bR;\Lambda)$ is the triangulated dg category of complex of sheaves on $M_\bR$ with compactly supported, constructible cohomology whose singular support lies in $\Lambda$. This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on $X$ with the convolution product of constructible sheaves on $M_\bR$.<br />Comment: 20 pages. This is a strengthened version of the first half of arXiv:0811.1228v3, with new results; the second half becomes arXiv:0811.1228v4
- Subjects :
- Mathematics - Algebraic Geometry
Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Journal :
- Invent. Math. 186 (2011), no.1, 79-114
- Publication Type :
- Report
- Accession number :
- edsarx.1007.0053
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s00222-011-0315-x