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P=W in composite rank for fixed determinant

OpenAI · September 24, 2026

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Abstract

We prove the P = W conjecture on the full rational cohomology of fixed-determinant, trace-free Higgs moduli spaces in composite rank and coprime degree, for smooth projective complex curves of genus at least two. Together with the established prime-rank cases, this gives the equality in every coprime rank.

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Possibly relevant literature (found by the agent, not checked)

Xueqing Wen, Multiplicativity of the perverse filtration in the SL_n case (arXiv:2610.03096, 2 October 2026). Studies the perverse filtration for the canonical SL_n Hitchin system with coprime rank and degree, using the endoscopic correspondence and variant cohomology; this bears on the paper's perverse-filtration analysis of the full fixed-determinant cohomology, including the variant part.
Anne Larsen, Fourier duality on compactified Prym fibrations (arXiv:2609.31604, 25 September 2026). Proves multiplicativity of the perverse filtration for compactified Prym fibrations, including SL Higgs moduli over the elliptic locus, extending the Maulik–Shen–Yin Fourier framework that the paper cites as [MSY2025].
Michael Groechenig, Dimitri Wyss, Paul Ziegler, Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration (arXiv:1707.06417, 2017). Proves the Hausel–Thaddeus topological mirror symmetry conjecture for smooth SL_n/PGL_n Higgs moduli in all ranks; this concerns the variant cohomology, which Theorem 1.1 newly covers in composite rank.
Camilla Felisetti, Mirko Mauri, P=W conjectures for character varieties with symplectic resolution (arXiv:2006.08752, 2020). Proves P=W and PI=WI for GL_n and SL_n character varieties that admit symplectic resolutions; this is a non-coprime companion to the paper's coprime fixed-determinant statement.
Junhui Qin, Moduli of twisted Higgs bundles (Oberwolfach Report 16/2024, pp. 974–978, workshop 31 March–5 April 2024). Section 2 reviews SLₙ endoscopic decomposition, including the canonical twist, providing background for the manuscript's extension of the filtration comparison to variant cohomology in composite rank.

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 043

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Cite

@misc{OAI:P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026,
  author = {{OpenAI}},
  title = {{$P=W$ in composite rank for fixed determinant}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026/P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026.pdf}{OAI:P-equals-W-in-composite-rank-for-fixed-determinant-September-24-2026}},
  year = {2026}
}