A categorification of Morelli's theorem

作者：Fang Bohan; Liu Chiu Chu Melissa^*; Treumann David; Zaslow Eric

来源：Inventiones Mathematicae, 2011, 186(1): 79-114.

DOI：10.1007/s00222-011-0315-x

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(R) = Lie(T(R)(boolean OR)) congruent to R(n) be the Lie algebra of the compact dual (real) torus T(R)(boolean OR) congruent to U(1)(n). Then there is a corresponding conical Lagrangian Lambda subset of T*M(R) and an equivalence of triangulated dg categories Perf(T) (X) congruent to Sh(cc)(M(R); Lambda), where Perf(T) (X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Sh(cc)(M(R); Lambda) is the triangulated dg category of complex of sheaves on M(R) 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(R).

• 出版日期2011-10
• 单位西北大学