Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces
Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces
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DOI：
10.1353/ajm.2012.0039
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发表时间：
2009-10
影响因子：
1.7
通讯作者：
Si Li;B. Lian;S. Yau
中科院分区：
文献类型：
--
作者：
Si Li;B. Lian;S. Yau
研究了光滑射影超曲面及其代数子曲面对的相对上同调变率。将Picard-Fuchs算子应用于环变Calabi-Yau超曲面上的全纯顶形式，构造了一个非齐次Picard-Fuchs方程，并在方程的一侧导出了$d$精确形式的一般公式。我们还推导了一个二重剩余公式，给出了一种纯代数的方法来计算Abel-Jacobi映射的非齐次Picard-Fuchs方程，这在最近的d膜研究中发挥了重要作用（由Morrison和Walcher）。利用变分形式证明了Calabi-Yau超曲面上的环面b膜的相对周期满足物理文献（Alim, Hecht, Mayr，和Mertens）提出的增强gkz -超几何系统，并讨论了近年来开放弦镜面对称研究中几项工作之间的关系。并给出了增强超几何系统的一般解。
We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.