A non-holomorphic P=W phenomenon
arXiv:2607.26806
The paper proves the P=W identity for 3‑dimensional isolated cluster varieties without the full rank assumption, using real‑analytic geometry to construct perverse truncations and a non‑proper algebraic morphism to analyze weight filtrations via mixed Hodge modules.
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.
46 pages. Comments are welcome!