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arXiv · 2607.26806

A non-holomorphic P=W phenomenon

Zili Zhang
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Abstract

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.

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Zili Zhang. 2026-07-29. A non-holomorphic P=W phenomenon. https://arxiv.org/abs/2607.26806

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