AI 中文总结
针对三维孤立簇簇代数簇，在不满足满秩假设的情况下证明P=W恒等式，研究其非全纯特性，分析拉格朗日纤维化的实解析几何与混合霍奇模导出范畴中常层的导出前推分解，揭示该情形下两类莱夫谢茨性质失效的现象。
AI 中文摘要
我们在不满足满秩假设的条件下，证明了三维孤立簇簇代数簇的P=W恒等式。这类簇代数簇通常是奇异的，且相关的拉格朗日纤维化并非复代数的。在P侧，我们通过对拉格朗日纤维化的显式实解析几何进行详细分析，构造了佩尔弗斯截断。在W侧，我们从簇代数簇出发构造了一个自然的非真代数态射，并在混合霍奇模的导出范畴内研究了沿该态射的常层的导出前推的分解。在三维非满秩孤立簇簇代数簇的P=W现象中，W侧的奇异硬莱夫谢茨性质和P侧的相对硬莱夫谢茨性质均不成立。
英文摘要
We prove the P=W identity for isolated cluster varieties of dimension 3 without the full rank hypothesis. These cluster varieties are generally singular, and the associated Lagrangian fibrations are not complex algebraic. On the P side, we construct the perverse truncation by a detailed analysis of the explicit real-analytic geometry of the Lagrangian fibration. On the W side, we construct a natural non-proper algebraic morphism from the cluster varieties and investigate the decomposition of the derived push-forward of the constant sheaf along this morphism within the derived category of mixed Hodge modules. In the P=W phenomenon for non-full rank isolated cluster varieties of dimension 3, both the curious hard Lefschetz property on the W side and the relative hard Lefschetz property on the P side fail.
Comments46 pages. Comments are welcome!
