arXiv 0811.12282008-11math.AGmath.SG

环面簇的T-对偶与同调镜像对称性

T-Duality and Homological Mirror Symmetry of Toric Varieties

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

备注：: 34 pages, 2 figures. The previous version of this paper has now been broken into two parts. The other part is available at arXiv:1007.0053
发表信息：: Adv. Math. 229 (2012), 1873-1911

摘要：

设$X_Σ$是一个完全复曲面簇。\cite{FLTZ}的相干可构造对应$κ$将$\Perf_T(X_Σ)$等同于向量空间$M_\bR.$上的可构造层的子范畴$Sh_{cc}(M_\bR;\LS)$。\cite{NZ,N}的微局部化等价$μ$将这些层与余切$T^*M_\bR$的Rakaya范畴的子范畴$Fuk(T^*M_\bR;\LS)$联系起来。当$X_\Si$是非奇异的，采取派生范畴产生同调镜像对称，$DCoh_T(X_\Si)\cong DFuk(T^*M_\bR;\LS)$，这是一个等价的三角张量范畴的等变版本。\cite{T}的非等变相干可构造对应$\barκ$将$\Perf(X_\Si)$嵌入到紧环面$T_\bR^\vee$上的可构造层的子范畴$Sh_c(T_\bR^\vee;\barΛ_\Si)$中。当$X_\Si$是非奇异的，$\barκ$和微局部化的组合产生一个同调镜像对称的版本，$DCoh(X_Σ)\hookrightarrow DFuk(T^*T_\bR;\barΛ_\Si)$，这是一个完整的嵌入三角张量范畴。当$X_\Si$是非奇异的和投射的，组合$τ=μ\circ κ$与T-对偶相容，在下面的意义上。一个等变的样本线丛$\cL$在实环面下有一个厄米度量不变量，它的连接定义了实环面轨道上的一族平坦线丛。这个数据产生了一个T-对偶拉格朗日膜$\mathbb L$上的通用覆盖$T^*M_\bR$的双重真正的环面纤维化。证明了$\mathbb L\cong τ(\cL)$在$Fuk(T^*M_\bR;\LS).$中的存在性。

英文摘要：

Let $X_Σ$ be a complete toric variety. The coherent-constructible correspondence $κ$ of \cite{FLTZ} equates $\Perf_T(X_Σ)$ with a subcategory $Sh_{cc}(M_\bR;\LS)$ of constructible sheaves on a vector space $M_\bR.$ The microlocalization equivalence $μ$ 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 $\barκ$ of \cite{T} embeds $\Perf(X_\Si)$ into a subcategory $Sh_c(T_\bR^\vee;\barΛ_\Si)$ of constructible sheaves on a compact torus $T_\bR^\vee$. When $X_\Si$ is nonsingular, the composition of $\barκ$ and microlocalization yields a version of homological mirror symmetry, $DCoh(X_Σ)\hookrightarrow DFuk(T^*T_\bR;\barΛ_\Si)$, which is a full embedding of triangulated tensor categories. When $X_\Si$ is nonsingular and projective, the composition $τ=μ\circ κ$ 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 $\mathbb L\cong τ(\cL)$ in $Fuk(T^*M_\bR;\LS).$ Thus, equivariant homological mirror symmetry is determined by T-duality.

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