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de Cataldo–Hausel–Migliorini P=W conjecture

Let CC be a smooth projective curve, and let MBM_B be the Betti moduli space of (twisted) GL(n,C)GL(n,\mathbb{C})-representations of π1(C)\pi_1(C) and MDM_D the Dolbeault moduli space of semistable Higgs bundles of degree dd and rank nn. Let π:MD→A\pi:M_D\to\mathbb{A} be the proper Hitchin map. Simpson's canonical diffeomorphism identifies their cohomology groups:

H∗(MD,Q)=H∗(MB,Q).H^*(M_D,\mathbb{Q})=H^*(M_B,\mathbb{Q}).

de Cataldo–Hausel–Migliorini P=W conjecture. Under this identification, the perverse filtration associated with π\pi satisfies

PkH∗(MD,Q)=W2kH∗(MB,Q)=W2k+1H∗(MB,Q),k≥0.P_k H^*(M_D,\mathbb{Q})=W_{2k}H^*(M_B,\mathbb{Q})=W_{2k+1}H^*(M_B,\mathbb{Q}),\qquad k\geq 0.

This conjecture predicts that the perverse filtration on the Dolbeault moduli space matches the mixed Hodge-theoretic weight filtration on the Betti moduli space. The source presents it as a conjecture; no resolution status is supplied here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. de Cataldo–Hausel–Migliorini P=W conjecture

    Let h:M→Ah:{\mathcal M}\to{\mathcal A} be the Hitchin fibration, where M{\mathcal M} is either Mn,L{\mathcal M}_{n,L} or M~n,d\widetilde{{\mathcal M}}_{n,d}. Let MB{\mathcal M}^{B} be the associated Betti moduli space, and use non-abelian Hodge theory to identify H∗(M,C)H^*({\mathcal M},{\mathbb C}) with H∗(MB,C)H^*({\mathcal M}^{B},{\mathbb C}). Denote by PkP_k the perverse filtration induced by hh and by W2kW_{2k} the weight filtration on the mixed Hodge structure of MB{\mathcal M}^{B}. P=W conjecture. Under the non-abelian Hodge correspondence,

    PkHi(M,C)=W2kHi(MB,C).P_kH^i({\mathcal M},{\mathbb C})=W_{2k}H^i({\mathcal M}^{B},{\mathbb C}).

    The conjecture links the perverse filtration of the Hitchin fibration with the weight filtration on the corresponding Betti moduli space, and is a central cohomological question in non-abelian Hodge theory. The supplied text gives no resolution status.

    source: Davesh Maulik and Junliang Shen, “Endoscopic decompositions and the Hausel-Thaddeus conjecture”, arXiv:2008.08520 (2025).

Sources & referencesView supporting material

Primary source

Zili Zhang, “Perverse filtration for generalized Kummer varieties of fibered surfaces”, arXiv:2206.09587 (2025).

hitchin-fibrationshitchin-systemsmixed-hodge-structuresnonabelian-hodge-theoryperverse-filtrations

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