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Geometric P=W conjecture for character varieties

From papers

Let MDol(S,C⋅){\rm M}_{Dol}(S,C_{\cdot}) be the Dolbeault (Hitchin) moduli space and MB(S,C⋅){\rm M}_B(S,C_{\cdot}) the Betti character variety. Let NDol∗N^{\ast}_{Dol} and NB∗N^{\ast}_B be neighborhoods of infinity in these spaces, and let nn be half the complex dimension of the moduli space. The Hitchin map gives a map NDol∗→S2n−1N^{\ast}_{Dol}\to S^{2n-1}, while the dual boundary construction gives a map NB∗→D∂MB(S,C⋅)N^{\ast}_B\to {{\mathbb D}\partial}{\rm M}_B(S,C_{\cdot}). Geometric P=W conjecture. There is a homotopy-commutative square

NDol∗→∼NB∗↓↓S2n−1→∼D∂MB(S,C⋅)\begin{array}{ccc} N^{\ast}_{Dol}&\stackrel{\sim}{\rightarrow}&N^{\ast}_B\\ \downarrow&&\downarrow\\ S^{2n-1}&\stackrel{\sim}{\rightarrow}&{{\mathbb D}\partial}{\rm M}_B(S,C_{\cdot}) \end{array}

where the top and side maps are the maps described above, and the bottom map is a homotopy equivalence. This conjecture proposes a geometric identification between the sphere at infinity of the Hitchin moduli space and the dual boundary complex of the Betti moduli space. The paper proves the existence of a homotopy equivalence of the asserted type for rank-two local systems on a punctured projective line, but does not establish homotopy commutativity of the diagram.

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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. The geometric P=W conjecture for character varieties

    Let GG be a reductive Lie group, let gg be the genus of a closed Riemann surface, and let Xg(G)\mathcal{X}_{g}(G) denote the corresponding character variety. A dlt log Calabi–Yau compactification is a compactification whose boundary pair is divisorial log terminal and log Calabi–Yau. The dual intersection complex of a boundary divisor records the incidence relations among its irreducible components. Geometric P=W conjecture. Xg(G)\mathcal{X}_{g}(G) admits a dlt log Calabi–Yau compactification, and the dual intersection complex of the boundary divisor is a polyhedral complex homeomorphic to a sphere. This is a geometric refinement of the P=W framework, relating the asymptotic behavior of the non-abelian Hodge correspondence to compactifications of character varieties and Higgs-bundle moduli spaces. The source presents it as a conjecture; no resolution is supplied here.

    source: Ashwin Ayilliath-Kutteri, Mohammad Farajzadeh-Tehrani and Charles Frohman, “The Geometric P=W conjecture and Thurston's compactification”, arXiv:2507.07211 (2026).

Sources & referencesView supporting material

Primary source

Carlos Simpson, “The dual boundary complex of the SL_2 character variety of a punctured sphere”, arXiv:1504.05395 (2015).

character-varietiescompactificationsdual-complexeshitchin-fibrationsnonabelian-hodge-theory

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