Algebraic and complex geometry · Result 043

P = W for fixed-determinant SL_n moduli spaces

1 paper

Proves \(P_k=W_{2k}=W_{2k+1}\) 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.

Watch

▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 1

P=W in composite rank for fixed determinant

OpenAI · September 24, 2026

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.

Read PDF Download Source

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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.