Geometric P=W Conjecture Overview

Updated 6 July 2026

The geometric P=W conjecture is a topological refinement of non-abelian Hodge theory that compares the asymptotic geometry of the Dolbeault and Betti moduli spaces.
It establishes a homotopy equivalence between the Hitchin sphere and the dual boundary complex, replacing filtration equalities with compatibility at infinity.
Recent proofs in rank one, genus one, and Painlevé cases employ methods like dlt compactifications, plumbing calculus, and spectral abelianization.

Searching arXiv for recent and foundational papers on the geometric $P=W$ conjecture. The geometric $P=W$ conjecture is a topological and birational refinement of the cohomological $P=W$ conjecture in non-abelian Hodge theory. In its basic form, it compares the asymptotic geometry of the Dolbeault moduli space, seen through the Hitchin fibration, with the asymptotic geometry of the Betti moduli space, seen through a compactification and its dual boundary complex. For singular character varieties, one formulation asks for a homotopy equivalence

$r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$

such that, for suitable neighborhoods of infinity, the diagram relating the Hitchin map, the non-abelian Hodge correspondence, and the evaluation map to the dual complex is homotopy commutative (Mauri et al., 2018). In this sense, the conjecture replaces an equality of filtrations on cohomology by a compatibility statement at infinity between the Hitchin sphere and the boundary complex.

1. Formulation and conceptual position

The cohomological $P=W$ conjecture of de Cataldo–Hausel–Migliorini compares the perverse filtration on the Dolbeault side with the weight filtration on the Betti side. The geometric $P=W$ conjecture, introduced in the work of Katzarkov–Noll–Pandit–Simpson and developed for singular character varieties in (Mauri et al., 2018), is a stronger statement: it predicts that the boundary behavior of the two spaces is compatible already at the level of homotopy.

For a compact Riemann surface $C$ and a complex reductive group $G$ , the two moduli spaces are the Dolbeault moduli space $M_{\mathrm{Dol}}(C,G)$ and the Betti moduli space

$M_{\mathrm B}(C,G)=\operatorname{Hom}(\pi_1(C),G)\sslash G.$

They are related by the non-abelian Hodge correspondence

$P=W$ 0

which is real analytic but not algebraic (Mauri et al., 2018).

On the Dolbeault side, the Hitchin map

$P=W$ 1

is a proper algebraic fibration. Choosing a semialgebraic neighborhood $P=W$ 2 of infinity yields a map to the sphere at infinity,

$P=W$ 3

On the Betti side, one considers a dlt compactification $P=W$ 4 and the dual boundary complex $P=W$ 5, together with an evaluation map

$P=W$ 6

constructed from a partition of unity subordinate to neighborhoods of boundary divisors (Mauri et al., 2018).

In this language, geometric $P=W$ 7 predicts the existence of a homotopy equivalence $P=W$ 8 making the square

$P=W$ 9

homotopy commutative (Mauri et al., 2018). The abridged slogan in that paper is

$P=W$ 0

but the full conjecture includes the compatibility diagram, not only the sphere topology.

2. Boundary geometry, compactifications, and dual complexes

The Betti side of geometric $P=W$ 1 is governed by compactification theory. For singular character varieties, the appropriate framework is that of dlt compactifications, because snc compactifications need not exist (Mauri et al., 2018). The resulting dual complex is defined combinatorially from the strata of the boundary divisor, and its topology is stable up to $P=W$ 2-homeomorphism under dlt modification.

A key structural result is the identification of dual complexes with non-Archimedean skeletons: $P=W$ 3 together with a quotient formula

$P=W$ 4

for finite quotients (Mauri et al., 2018). These statements make the dual boundary complex computable in settings built from symmetric products or finite-group quotients.

For genus-one character varieties, the dual complexes were computed explicitly: $P=W$ 5 (Mauri et al., 2018). These are among the first higher-rank realizations of the conjectural sphere at infinity.

A closely related recent construction concerns the closed-surface $P=W$ 6-character variety $P=W$ 7. In that setting, the conjecture is phrased as the existence of a dlt log CY compactification whose dual intersection complex is a polyhedral complex homeomorphic to a sphere, and a projective compactification is constructed whose boundary divisors are toric orbifolds and whose dual intersection complex is a sphere (Ayilliath-Kutteri et al., 9 Jul 2025). That paper states that the compactification may not be dlt without further toric blowups, and that the log CY property remains open in the closed case (Ayilliath-Kutteri et al., 9 Jul 2025).

3. Relation to the cohomological $P=W$ 8 conjecture

The geometric conjecture was formulated partly to explain the cohomological $P=W$ 9 relation. In (Mauri et al., 2018), the cohomological statement is written for intersection cohomology as

$r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 0

The same paper proves that geometric $r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 1 implies the highest-weight part of this equality under a generic smoothness assumption: $r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 2 (Mauri et al., 2018).

The bridge between topology at infinity and top weight on cohomology is the standard identification

$r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 3

for a dlt compactification of an $r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 4-dimensional variety (Mauri et al., 2018). Thus the homotopy type of the dual complex controls the top graded piece of the weight filtration.

This relation is visible in low-dimensional wild character varieties as well. The Painlevé analysis in (Némethi et al., 2020) emphasizes that the geometric statement is expected to explain the original $r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 5 conjecture, at least on the lowest-weight graded piece. The paper also records Harder’s observation that, via Auroux’s conjectural framework, the geometric statement implies the lowest weight part of the classical $r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 6 relation (Némethi et al., 2020).

In the abelian-variety setting, the connection becomes completely explicit. The paper on $r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 7 phenomena on abelian varieties proves both a cohomological statement

$r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 8

and a geometric statement identifying neighborhoods of infinity through a commuting diagram involving

$r:\mathbb{S}^{N-1}\to \mathcal{D}(\partial M_{\mathrm B})$ 9

(Bolognese et al., 2023).

4. Established cases and concrete realizations

Several cases of the geometric conjecture, or of its homotopy-commutativity assertion, have been proved.

Setting	Result	Source
Rank one, arbitrary genus	Geometric $P=W$ 0 proved	(Mauri et al., 2018)
Genus one, $P=W$ 1	Geometric $P=W$ 2 proved	(Mauri et al., 2018)
Painlevé moduli spaces	Homotopy commutativity proved in all Painlevé cases	(Némethi et al., 2020)
Painlevé VI	Homotopy commutativity proved via abelianization near infinity	(Szabo, 2019)
Complex abelian varieties, arbitrary rank $P=W$ 3	Full analogue of both cohomological and geometric $P=W$ 4 proved	(Bolognese et al., 2023)

For Painlevé moduli spaces, (Némethi et al., 2020) considers

$P=W$ 5

and proves that, for sufficiently large $P=W$ 6, there is a homotopy commutative square

$P=W$ 7

with $P=W$ 8 (Némethi et al., 2020). The same paper shows that the Dolbeault and Betti boundary $P=W$ 9-manifolds are orientation-preserving diffeomorphic, while their mixed Hodge structures differ in the manner predicted by $P=W$ 0 (Némethi et al., 2020).

For Painlevé VI, (Szabo, 2019) gives a different proof based on abelianization of Higgs bundles near infinity. In that case, the dual complex $P=W$ 1 of the boundary divisor in a smooth compactification of the cubic surface character variety is homeomorphic to $P=W$ 2, and the theorem shows that the boundary loop coming from the Hitchin fibration maps to a generator of

$P=W$ 3

(Szabo, 2019).

For complex abelian varieties $P=W$ 4, the geometric statement uses the spectral data morphism

$P=W$ 5

and proves the existence of neighborhoods of infinity

$P=W$ 6

such that the non-abelian Hodge homeomorphism fits into a commuting diagram with radial retraction

$P=W$ 7

and a retraction to the dual complex of a dlt compactification on the Betti side (Bolognese et al., 2023).

5. Methods of proof

The methods used in geometric $P=W$ 8 are notably diverse, because the conjecture concerns topology at infinity rather than only cohomology.

In the birational and non-Archimedean approach, the basic tools are dlt compactifications, skeletons, the minimal model program, quotient formulas for dual complexes, and degeneration arguments (Mauri et al., 2018). The same paper gives alternative proofs of genus-one results via degenerations of Hilbert schemes of points on K3 surfaces and generalized Kummer varieties (Mauri et al., 2018).

For Painlevé spaces, (Némethi et al., 2020) uses plumbing calculus. Both the Dolbeault and Betti boundary $P=W$ 9-manifolds are described by plumbing graphs, and Neumann’s moves $C$ 0, $C$ 1, and $C$ 2 are used to prove that the two plumbed $C$ 3-manifolds are orientation-preserving diffeomorphic (Némethi et al., 2020). The homotopy commutativity then reduces to a computation in

$C$ 4

for connected oriented $C$ 5-manifolds with $C$ 6 (Némethi et al., 2020).

A different analytic strategy appears in the Painlevé VI case. The spectral curve is a double cover

$C$ 7

branched at $C$ 8, and topologically $C$ 9 is a torus (Szabo, 2019). Mochizuki’s asymptotic theorem provides local frames in which the Higgs field becomes asymptotically diagonal near infinity, and the resulting monodromy asymptotics on the Betti side reduce to period integrals of the spectral differential (Szabo, 2019). This converts the boundary problem into a problem about dominant asymptotics of trace coordinates on a cubic surface compactification.

The rank- $G$ 0 five-punctured sphere case also uses asymptotic abelianization away from the ramification divisor, together with fiducial solutions near punctures and branch points, to identify torus cycles in generic Hitchin fibers with boundary tori in a compactification of the Betti moduli space (Szabo, 2021). That paper proves the lowest weighted piece of $G$ 1, rather than the full geometric conjecture, but it exhibits the same mechanism: generic Hitchin fibers control low perverse degree, while boundary tori control low weight (Szabo, 2021).

In the abelian-variety case, the method is completely different. Because every stable topologically trivial Higgs bundle has rank one, the Dolbeault moduli space is

$G$ 2

the Betti moduli space is

$G$ 3

and the boundary at infinity is controlled by the symmetric product $G$ 4 rather than by the usual affine Hitchin base (Bolognese et al., 2023). The geometry is therefore explicit from the outset.

The geometric $G$ 5 conjecture has generated several extensions. One direction concerns singular and stacky moduli. The stacky $G$ 6 framework compares weight filtrations on Betti Borel–Moore homology with a “less perverse filtration” on Dolbeault Borel–Moore homology and proves the resulting conjectures in genus $G$ 7 and $G$ 8 (Davison, 2021). This is not the same statement as geometric $G$ 9, but it is part of the broader attempt to extend the $M_{\mathrm{Dol}}(C,G)$ 0 philosophy beyond smooth coarse moduli spaces.

Another direction is mirror symmetry. The paper “ $M_{\mathrm{Dol}}(C,G)$ 1 Phenomena” formulates a mirror $M_{\mathrm{Dol}}(C,G)$ 2 conjecture for log Calabi–Yau mirror pairs, where the weight filtration on one side is expected to match the perverse Leray filtration of the affinization map on the mirror side (Harder et al., 2019). In the Fano/Landau–Ginzburg setting, hybrid LG models were introduced to formulate a multi-potential mirror $M_{\mathrm{Dol}}(C,G)$ 3 conjecture for simple normal crossing anticanonical divisors (Lee, 2022), and in the semi-stable degeneration setting the monodromy weight filtration is predicted to correspond to the perverse Leray filtration of a mirror fibration over $M_{\mathrm{Dol}}(C,G)$ 4 (Lee, 2023). These mirror-theoretic variants preserve the central theme of geometric $M_{\mathrm{Dol}}(C,G)$ 5: boundary geometry on one side corresponds to perverse geometry on the other.

Several major problems remain open. For higher genus and higher rank, geometric $M_{\mathrm{Dol}}(C,G)$ 6 is still open in general (Mauri et al., 2018). It is also open whether $M_{\mathrm{Dol}}(C,G)$ 7 admits a dlt log Calabi–Yau compactification in all genera $M_{\mathrm{Dol}}(C,G)$ 8 (Mauri et al., 2018). Even when the dual boundary complex is known or conjecturally sphere-like, the current geometric formalism only yields the top-weight part of cohomological $M_{\mathrm{Dol}}(C,G)$ 9; a full geometric mechanism for the lower perverse and lower weight pieces has not yet been extracted from the sphere-at-infinity picture (Mauri et al., 2018). A plausible implication is that further progress will require a refinement of the conjecture beyond the present dual-complex formulation.