Result 043, Algebraic and complex geometry

P = W for fixed-determinant SLn moduli spaces

Proves Pk=W2k=W2k+1P_k=W_{2k}=W_{2k+1} on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.

Proof

The bigger picture

Why it matters

Two ways of organizing the topology of spaces associated with complex curves are claimed to agree. This connects geometric information from Higgs fields with algebraic information from representations of the curve's fundamental group.

What changes?

The manuscript reports the equality on full rational cohomology, including variant cohomology, for smooth fixed-determinant, trace-free Higgs moduli spaces and their corresponding character varieties. These spaces classify bundles equipped with Higgs fields and representations, respectively. The assumptions are composite rank, degree coprime to rank, and a smooth projective complex curve of genus at least two. Precisely, perverse level k equals weight levels 2k and 2k+1. Combined with established prime-rank cases, this covers every coprime rank.

What does that help mathematicians do?

The perverse filtration organizes cohomology through the Hitchin map, which records characteristic data of Higgs fields. The weight filtration organizes it by algebraic complexity on the character-variety side. Their equality lets researchers translate information between these descriptions, including the variant part rather than only a restricted collection of classes. It also rules out new classes entering at odd weight levels, without asserting that odd-degree cohomology vanishes.

Are there practical applications?

The immediate value is foundational: the claimed correspondence makes two different descriptions of these moduli spaces usable together when studying their topology. It extends that comparison to composite ranks under the stated coprimality assumption. The supplied sources describe no practical deployment or computational performance improvement; the consequence concerns the mathematical structure of these spaces.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

P=W in composite rank for fixed determinant

September 24, 2026 39 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

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