In this note, we study homology classes in the mirror quintic Calabi–Yau threefold that can be realized by special Lagrangian submanifolds. We have used Picard–Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers $p,$ we can compute the orbit modulo p. We conjecture that the orbit in homology with coefficients in $\mathbb {Z}$ can be determined by these orbits with coefficients in $\mathbb {Z}_p$ .

中文翻译：

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镜像五次三重中拉格朗日子流形支持的同源性

在本笔记中，我们研究了可以通过特殊拉格朗日子流形实现的镜像五次 Calabi-Yau 三重中的同源类。我们已经使用 Picard-Lefschetz 理论来建立单调作用并研究拉格朗日消失循环的轨道。对于许多质数 $p,$ 我们可以计算轨道模p。我们推测系数在 $\mathbb {Z}$ 的 轨道可以由这些系数在 $\mathbb {Z}_p$ 的 轨道确定 。