$\def\M{\mathscr{M}}\def\Rscr{\mathscr{R}}\def\Rsf{\mathsf{R}}\def\Tsf{\mathsf{T}}\def\tildeM{\widetilde{\M}}$For any positive integer $n$, we introduce a modular vector field $\Rsf$ on a moduli space $\Tsf$ of enhanced Calabi–Yau $n$-folds arising from the Dwork family. By Calabi–Yau quasi-modular forms associated to $\Rsf$ we mean the elements of the graded $\mathbb{C}$-algebra $\tildeM$ generated by solutions of $\Rsf$, which are provided with natural weights. The modular vector field $\Rsf$ induces the derivation $\Rscr$ and the Ramanujan–Serre type derivation $\partial$ on $\tildeM$. We show that they are degree $2$ differential operators and there exists a proper subspace $\M \subset \tildeM$, called the space of Calabi–Yau modular forms associated to $\Rsf$, which is closed under $\partial$. Using the derivation $\Rscr$, we define the Rankin–Cohen brackets for $\tildeM$ and prove that the subspace generated by the positive weight elements of $\M$ is closed under the Rankin–Cohen brackets. We find the mirror map of the Dwork family in terms of the Calabi–Yau modular forms.

中文翻译：

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Calabi-Yau 模块化形式的 Rankin-Cohen 支架


$\def\M{\mathscr{M}}\def\Rscr{\mathscr{R}}\def\Rsf{\mathsf{R}}\def\Tsf{\mathsf{T}}\def\tildeM{ \widetilde{\M}}$对于任何正整数$n$，我们在源自 Dwork 系列的增强 Calabi-Yau $n$ 折​​叠的模空间 $\Tsf$ 上引入模向量场 $\Rsf$。与 $\Rsf$ 相关的 Calabi-Yau 准模形式是指由 $\Rsf$ 的解生成的分级 $\mathbb{C}$-代数 $\tildeM$ 的元素，这些元素具有自然权重。模向量场 $\Rsf$ 在 $\tildeM$ 上推导 $\Rscr$ 和 Ramanujan-Serre 型推导 $\partial$。我们证明它们是$2$度微分算子，并且存在一个真子空间$\M \subset \tildeM$，称为与$\Rsf$相关的Calabi-Yau模形式空间，它在$\partial$下是封闭的。利用推导$\Rscr$，我们定义了$\tildeM$的Rankin-Cohen括号，并证明$\M$的正权重元素生成的子空间在Rankin-Cohen括号下是封闭的。我们以卡拉比-丘模形式的形式找到了德沃克家族的镜像映射。