Let M_Dol be the Dolbeault moduli space and M_B the corresponding Betti character variety. Let P_bullet be the perverse filtration on H^(M_Dol) induced by the proper Hitchin map, and let W_bullet be the weight filtration on H^(M_B). P=W conjecture. Under the non-abelian Hodge isomorphism H^(M_Dol)cong H^(M_B), P_k(M_Dol)cong W_2k(M_B). The conjecture explains the curious hard Lefschetz symmetry through relative hard Lefschetz for the Hitchin fibration. The source states that it was proved for G=operatornameGL_2,operatornamePGL_2,operatornameSL_2; no general resolution is supplied here.

P=W conjecture for the Hitchin system

Let $\mathcal{M}_\Dol$ be the Dolbeault moduli space and $\mathcal{M}_\B$ the corresponding Betti character variety. Let $P_\bullet$ be the perverse filtration on $H^*(\mathcal{M}_\Dol)$ induced by the proper Hitchin map, and let $W_\bullet$ be the weight filtration on $H^*(\mathcal{M}_\B)$ . P=W conjecture. Under the non-abelian Hodge isomorphism $H^*(\mathcal{M}_\Dol)\cong H^*(\mathcal{M}_\B)$ ,

P_k(\mathcal{M}_\Dol)\cong W_{2k}(\mathcal{M}_\B).

The conjecture explains the curious hard Lefschetz symmetry through relative hard Lefschetz for the Hitchin fibration. The source states that it was proved for $G=\operatorname{GL}_2,\operatorname{PGL}_2,\operatorname{SL}_2$ ; no general resolution is supplied here.

Sources & referencesView supporting material

Primary source

Tamas Hausel, “Global topology of the Hitchin system”, arXiv:1102.1717 (2011).

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Progress summary

Refreshed 3 days ago

Partially solved

The conjecture is now proved for every general linear group, while versions for other groups and singular spaces remain open.

The $P=W$ conjecture identifies the perverse filtration from the Hitchin map with the weight filtration on the corresponding character variety under non-abelian Hodge theory. The broad conjecture is not fully settled across all reductive groups and moduli-space variants.

Known results

$\operatorname{GL}_2$ , $\operatorname{SL}_2$ , and $\operatorname{PGL}_2$ : de Cataldo–Hausel–Migliorini.
Arbitrary rank in genus $2$ : de Cataldo–Maulik–Shen for $\operatorname{GL}_n$ ; related prime-rank $\operatorname{SL}_p$ cases were also known.
The intersection-cohomology formulation remained conjectural for general complex reductive groups in the 2022 literature.

Full $\operatorname{GL}_n$ proof (date unavailable)

A paper proves $P=W$ for $\operatorname{GL}_n$ in every rank and every curve of genus $g\ge 2$ , using tautological-class perversity, vanishing cycles, global Springer theory, parabolic support, and Mellit’s curious hard Lefschetz theorem. A later source reports independent proofs, including work of Hausel–Mellit–Minets–Schiffmann.

Current status (as of August 2026): $P=W$ is settled for smooth $\operatorname{GL}_n$ moduli spaces in all ranks and genera $g\ge 2$ , with related type- $A$ consequences, but remains open in general reductive-group, singular, intersection-cohomology, stack, and parabolic formulations.

Sources

No solutions have been posted yet.