Mathematical Research Letters

Volume 28 (2021)

Number 3

The $P=W$ identity for cluster varieties

Pages: 925 – 944



Zili Zhang (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


We find new examples of the $P = W$ identities of de Cataldo–Hausel–Migliorini by studying cluster varieties. We prove that the weight filtration of 2D cluster varieties correspond to the perverse filtration of elliptic fibrations which are deformation equivalent to the elliptic fibrations of types $I_b$ or $2I_b$. These examples do not arise from character varieties or moduli of Higgs bundles.

Received 19 June 2019

Accepted 24 February 2020

Published 2 June 2021