We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these $\log$ Calabi–Yau varieties underlie a Landau–Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is $\log$ Calabi–Yau as it isomorphic to the complement of the Knutson–Lam–Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.

中文翻译：

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开理查森簇的基本群

我们在对应于部分标志流形的完整标志流形中计算开理查森簇的基本群。Rietsch 表明，这些 $\log$ Calabi–Yau 变体是 Langlands 双分旗流形的 Landau–Ginzburg 镜像的基础，我们的计算验证了该镜像的 Hori 预测。它是 $\log$ Calabi–Yau，因为它同构于部分标志流形的 Knutson–Lam–Speyer 反规范除数的补集。我们还确定了该除数的显式定义方程。