A categorification of Morelli’s theorem
A categorification of Morelli’s theorem
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莫雷利定理的分类
DOI：
10.1007/s00222-011-0315-x
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发表时间：
2010
影响因子：
3.1
通讯作者：
E. Zaslow
中科院分区：
文献类型：
--
作者：
Bohan Fang;Chiu;David Treumann;E. Zaslow
在向量空间中，证明了环变相干轴与多面体可构造轴之间的关系。在k理论水平上，该定理恢复了Morelli关于光滑投影环变的k理论的描述（Morelli in Adv. mathematics . 100(2): 154-182, 1993）。具体地说，设X是维数n的固有环面变分，设$M_{\mathbb{R}} = \mathrm{Lie}(T_{\mathbb{R}}^{\vee})\cong\mathbb {R}^{n}$是紧对偶（实）环面的李代数$T_{\mathbb{R}}^{\vee}\cong U(1)^{n}$。那么就有一个相应的圆锥拉格朗日Λ∧T * M∈和一个三角化dg范畴的等价$\mathcal{P}\mathrm{erf}_{T}(X) \cong\mathit{Sh}_{cc}(M_{\mathbb{R}};\Lambda)$，其中$\mathcal{P}\mathrm{erf}_{T}(X)$是X上环面等变相干轴的完美复形的三角化dg范畴，Shcc（M∈；Λ）是M∈上具有紧支撑的可构造上同调的轴的复形的三角化dg范畴，其奇异支撑位于Λ。这个等价是单一性的——它将X上相干束的张量积与M上可构造束的卷积积交织在一起。
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.