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The P = W conjecture for fixed-determinant (SL_n) Higgs moduli in composite rank

For a smooth projective curve CC of genus g≥2g\ge2 and gcd⁡(n,d)=1\gcd(n,d)=1, nonabelian Hodge theory identifies the cohomology of the moduli space MDolL(SLn)M^L_{\mathrm{Dol}}(\mathrm{SL}_n) of stable trace-free Higgs bundles with fixed determinant and that of the twisted SLn\mathrm{SL}_n character variety. De Cataldo, Hausel and Migliorini conjectured that the perverse filtration PP of the Hitchin map matches the weight filtration WW: Pk=W2k=W2k+1P_k=W_{2k}=W_{2k+1}. Maulik-Shen and others proved it for GLn\mathrm{GL}_n; for fixed determinant it was known in prime rank, where the variant cohomology was controlled. Does P=WP=W hold on the full cohomology, variant part included, for every composite rank nn?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Algebraic geometry; nonabelian Hodge theory, Higgs bundles
Posed by
M. A. de Cataldo, T. Hausel and L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties: the case A_1, Ann. of Math. 175 (2012)
Year posed
2012
Years open
14y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
32 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: Pk=W2k=W2k+1P_k=W_{2k}=W_{2k+1} on all of H∗(XB,Q)H^*(X^B,\mathbb Q) for the smooth coprime fixed-determinant, trace-free moduli in every composite rank n≥4n\ge4, genus g≥2g\ge2, including the variant (non-Γ\Gamma-invariant) cohomology; with the earlier prime-rank theorems this gives the fixed-determinant conjecture in every coprime rank. A corollary answers Maulik-Shen's Question 5.5 on Betti weights of endoscopic images. Non-coprime (singular) cases and other groups are not treated.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as the paper states it; it claims PkHm=W2kHm=W2k+1HmP_kH^m=W_{2k}H^m=W_{2k+1}H^m on the full rational cohomology for every composite n≥4n\ge4, coprime dd, every LL and every genus g≥2g\ge2 curve. The proof (an sl2\mathfrak{sl}_2 operator built from logarithmic moving-pole moduli, following Hausel-Mellit-Minets-Schiffmann) was not refereed and is not formalized. It uses Mellit's toric stratifications and published logarithmic Hodge moduli constructions as inputs.

Source

Changelog1 change

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