In this paper, we study the zero loci of local systems of the form $\delta\Pi$, where $\Pi$ is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space $X$, and $\delta$ is a given differential operator on the space of sections $V^\vee=\Gamma(X,K_X^{-1})$. Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of $\delta\Pi$. As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus.

中文翻译：

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周期积分和广义超几何函数的微分零点

在本文中，我们研究了 $\delta\Pi$ 形式的局部系统的零轨迹，其中 $\Pi$ 是在合适的环境空间 $X$ 中 CY 超曲面的通用族的周期层，并且 $\ delta$ 是 $V^\vee=\Gamma(X,K_X^{-1})$ 部分空间上的给定微分算子。使用三位作者及其合作者的早期结果，我们对 $\delta\Pi$ 的零位点给出了几种不同的描述。作为应用，我们证明轨迹是代数的，并且在某些情况下是非空的。我们还给出了在某些情况下计算轨迹的多项式定义方程的明确方法。这种描述导致了零位点的自然分层。