In this paper, we describe recent work towards the mirror P=W conjecture, which relates the weight filtration on a cohomology of a log Calabi--Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi--Yau manifold, taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi--Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini, and de Cataldo and Migliorini. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror P=W conjecture provides a possible bridge between the mirror P=W conjecture and the well-known P=W conjecture in nonabelian Hodge theory.

中文翻译：

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$$\mathrm P=\mathrm W$$ 现象

在本文中，我们描述了镜像 P=W 猜想的最新工作，该猜想将 log Calabi--Yau 流形的上同调上的权重过滤与同调镜像对数对数 Calabi--Yau 的上同调上的反常 Leray 过滤联系起来流形，相对于仿射图。这一猜想扩展了镜像对偶紧凑型 Calabi-Yau 流形的霍奇数之间的经典关系，并结合了 de Cataldo、Hausel 和 Migliorini 以及 de Cataldo 和 Migliorini 引人入胜的开创性作品中出现的工具和想法。我们对这个猜想的动机、最近的结果进行了广泛的概述，并描述了这个结果是如何从镜像对称的 SYZ 公式中产生的。