摘要翻译：
设$x_\sigma$是一个完全的toric变体。\cite{FLTZ}的相干可构造对应$\kappa$将$\perf_t(x_\sigma)$与向量空间$m_\br上的可构造束的子范畴$sh_{cc}(M_\br；\ls)$等价。$\cite{NZ,N}的微定位等价$\mu$将这些束与余切$t^*m_\br$的Fukaya范畴的子范畴$fuk(T^*m_\br；\ls)$相关。当$x_\si$非奇异时，取导出范畴得到同调镜像对称的等变形式$dcoh_t(x_\si)\cong DFuk(t^*m_\br；\ls)$,它是三角张量范畴的等价。cite{T}的非等变相干可构造对应$\bar{\kappa}$将$\perf(x_\si)$嵌入到紧环面上的可构造束的子类别$SH_c(T_\br^\vee；\bar{\lambda}_\si)$中$T_\br^\vee$。当$x_\si$非奇异时，$\bar{\kappa}$与微局域化的组合产生了一种同调镜像对称形式$dcoh(x_\sigma)\hookrightarrow DFuk(T^*t_\br；\bar{\lambda}_\si)$,它是三角张量范畴的完全嵌入。当$x_\si$是非奇异的射影时，合成$\tau=\mu\circ\kappa$在以下意义上与t-对偶相容。实环面上的等变丰满线丛$\cl$具有hermitian度量不变量，它的连接定义了实环面上的平坦线丛族。这个数据在对偶实环面纤维的通用覆盖$T^*M_\br$上得到了一个T-对偶拉格朗日膜$\MathBBL$。我们证明了$\mathbb L\cong\tau(\cl)$in$fuk(t^*m\br；\ls)$,从而由t-对偶决定了等变同调镜像对称性。
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英文标题：
《T-Duality and Homological Mirror Symmetry of Toric Varieties》
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作者：
Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, Eric Zaslow
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最新提交年份：
2010
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Symplectic Geometry        辛几何
分类描述：Hamiltonian systems, symplectic flows, classical integrable systems
哈密顿系统，辛流，经典可积系统
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英文摘要：
  Let $X_\Sigma$ be a complete toric variety. The coherent-constructible correspondence $\kappa$ of \cite{FLTZ} equates $\Perf_T(X_\Sigma)$ with a subcategory $Sh_{cc}(M_\bR;\LS)$ of constructible sheaves on a vector space $M_\bR.$ The microlocalization equivalence $\mu$ of \cite{NZ,N} relates these sheaves to a subcategory $Fuk(T^*M_\bR;\LS)$ of the Fukaya category of the cotangent $T^*M_\bR$. When $X_\Si$ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, $DCoh_T(X_\Si)\cong DFuk(T^*M_\bR;\LS)$, which is an equivalence of triangulated tensor categories.   The nonequivariant coherent-constructible correspondence $\bar{\kappa}$ of \cite{T} embeds $\Perf(X_\Si)$ into a subcategory $Sh_c(T_\bR^\vee;\bar{\Lambda}_\Si)$ of constructible sheaves on a compact torus $T_\bR^\vee$. When $X_\Si$ is nonsingular, the composition of $\bar{\kappa}$ and microlocalization yields a version of homological mirror symmetry, $DCoh(X_\Sigma)\hookrightarrow DFuk(T^*T_\bR;\bar{\Lambda}_\Si)$, which is a full embedding of triangulated tensor categories.   When $X_\Si$ is nonsingular and projective, the composition $\tau=\mu\circ \kappa$ is compatible with T-duality, in the following sense. An equivariant ample line bundle $\cL$ has a hermitian metric invariant under the real torus, whose connection defines a family of flat line bundles over the real torus orbits. This data produces a T-dual Lagrangian brane $\mathbb L$ on the universal cover $T^*M_\bR$ of the dual real torus fibration. We prove $\mathbb L\cong \tau(\cL)$ in $Fuk(T^*M_\bR;\LS).$ Thus, equivariant homological mirror symmetry is determined by T-duality.
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PDF链接：
https://arxiv.org/pdf/0811.1228